第1種スターリング数と第2種スターリング数の定義
第1種スターリング数と第2種スターリング数の定義
\(n\in\mathbb{N}_{0}\)とする。
\[ P\left(x,n\right)=\sum_{k=0}^{n}S_{1}\left(n,k\right)x^{k} \] \(n,k\in\mathbb{N}_{0}\text{で}\)\(n<0\lor k<0\)のときは一般的には定義されない。
\[ x^{n}=\sum_{k=0}^{n}S_{2}\left(n,k\right)P\left(x,k\right) \] \(n,k\in\mathbb{N}_{0}\text{で}\)\(n<0\lor k<0\)のときは一般的には定義されない。
\(n\in\mathbb{N}_{0}\)とする。
(1)第1種スターリング数
次の式で表される\(S_{1}\left(n,k\right)\)が第1種スターリング数である。\[ P\left(x,n\right)=\sum_{k=0}^{n}S_{1}\left(n,k\right)x^{k} \] \(n,k\in\mathbb{N}_{0}\text{で}\)\(n<0\lor k<0\)のときは一般的には定義されない。
(2)第2種スターリング数
次の式で表される\(S_{2}\left(n,k\right)\)が第2種スターリング数である。\[ x^{n}=\sum_{k=0}^{n}S_{2}\left(n,k\right)P\left(x,k\right) \] \(n,k\in\mathbb{N}_{0}\text{で}\)\(n<0\lor k<0\)のときは一般的には定義されない。
第1種スターリング数\(S_{1}\left(n,k\right)\)
\[ \begin{array}{|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|} \hline n\backslash k & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 & 13 & 14 & 15 & 16 & 17 & 18 & 19 & 20 & 21 & 22 & 23 & 24 & 25 & 26 & 27 & 28 & 29 & 30\\ \hline 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \hline 1 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \hline 2 & 0 & -1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \hline 3 & 0 & 2 & -3 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \hline 4 & 0 & -6 & 11 & -6 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \hline 5 & 0 & 24 & -50 & 35 & -10 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \hline 6 & 0 & -120 & 274 & -225 & 85 & -15 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \hline 7 & 0 & 720 & -1,764 & 1,624 & -735 & 175 & -21 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \hline 8 & 0 & -5,040 & 13,068 & -13,132 & 6,769 & -1,960 & 322 & -28 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \hline 9 & 0 & 40,320 & -109,584 & 118,124 & -67,284 & 22,449 & -4,536 & 546 & -36 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \hline 10 & 0 & -362,880 & 1,026,576 & -1,172,700 & 723,680 & -269,325 & 63,273 & -9,450 & 870 & -45 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \hline 11 & 0 & 3,628,800 & -10,628,640 & 12,753,576 & -8,409,500 & 3,416,930 & -902,055 & 157,773 & -18,150 & 1,320 & -55 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \hline 12 & 0 & -39,916,800 & 120,543,840 & -150,917,976 & 105,258,076 & -45,995,730 & 13,339,535 & -2,637,558 & 357,423 & -32,670 & 1,925 & -66 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \hline 13 & 0 & 479,001,600 & -1,486,442,880 & 1,931,559,552 & -1,414,014,888 & 657,206,836 & -206,070,150 & 44,990,231 & -6,926,634 & 749,463 & -55,770 & 2,717 & -78 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \hline 14 & 0 & -6,227,020,800 & 19,802,759,040 & -26,596,717,056 & 20,313,753,096 & -9,957,703,756 & 3,336,118,786 & -790,943,153 & 135,036,473 & -16,669,653 & 1,474,473 & -91,091 & 3,731 & -91 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \hline 15 & 0 & 87,178,291,200 & -283,465,647,360 & 392,156,797,824 & -310,989,260,400 & 159,721,605,680 & -56,663,366,760 & 14,409,322,928 & -2,681,453,775 & 368,411,615 & -37,312,275 & 2,749,747 & -143,325 & 5,005 & -105 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \hline 16 & 0 & -1,307,674,368,000 & 4,339,163,001,600 & -6,165,817,614,720 & 5,056,995,703,824 & -2,706,813,345,600 & 1,009,672,107,080 & -272,803,210,680 & 54,631,129,553 & -8,207,628,000 & 928,095,740 & -78,558,480 & 4,899,622 & -218,400 & 6,580 & -120 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \hline 17 & 0 & 20,922,789,888,000 & -70,734,282,393,600 & 102,992,244,837,120 & -87,077,748,875,904 & 48,366,009,233,424 & -18,861,567,058,880 & 5,374,523,477,960 & -1,146,901,283,528 & 185,953,177,553 & -23,057,159,840 & 2,185,031,420 & -156,952,432 & 8,394,022 & -323,680 & 8,500 & -136 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \hline 18 & 0 & -355,687,428,096,000 & 1,223,405,590,579,200 & -1,821,602,444,624,640 & 1,583,313,975,727,488 & -909,299,905,844,112 & 369,012,649,234,384 & -110,228,466,184,200 & 24,871,845,297,936 & -4,308,105,301,929 & 577,924,894,833 & -60,202,693,980 & 4,853,222,764 & -299,650,806 & 13,896,582 & -468,180 & 10,812 & -153 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \hline 19 & 0 & 6,402,373,705,728,000 & -22,376,988,058,521,600 & 34,012,249,593,822,720 & -30,321,254,007,719,424 & 17,950,712,280,921,504 & -7,551,527,592,063,024 & 2,353,125,040,549,984 & -557,921,681,547,048 & 102,417,740,732,658 & -14,710,753,408,923 & 1,661,573,386,473 & -147,560,703,732 & 10,246,937,272 & -549,789,282 & 22,323,822 & -662,796 & 13,566 & -171 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \hline 20 & 0 & -121,645,100,408,832,000 & 431,565,146,817,638,400 & -668,609,730,341,153,280 & 610,116,075,740,491,776 & -371,384,787,345,228,000 & 161,429,736,530,118,960 & -52,260,903,362,512,720 & 12,953,636,989,943,896 & -2,503,858,755,467,550 & 381,922,055,502,195 & -46,280,647,751,910 & 4,465,226,757,381 & -342,252,511,900 & 20,692,933,630 & -973,941,900 & 34,916,946 & -920,550 & 16,815 & -190 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \hline 21 & 0 & 2,432,902,008,176,640,000 & -8,752,948,036,761,600,000 & 13,803,759,753,640,704,000 & -12,870,931,245,150,988,800 & 8,037,811,822,645,051,776 & -3,599,979,517,947,607,200 & 1,206,647,803,780,373,360 & -311,333,643,161,390,640 & 63,030,812,099,294,896 & -10,142,299,865,511,450 & 1,307,535,010,540,395 & -135,585,182,899,530 & 11,310,276,995,381 & -756,111,184,500 & 40,171,771,630 & -1,672,280,820 & 53,327,946 & -1,256,850 & 20,615 & -210 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \hline 22 & 0 & -51,090,942,171,709,440,000 & 186,244,810,780,170,240,000 & -298,631,902,863,216,384,000 & 284,093,315,901,811,468,800 & -181,664,979,520,697,076,096 & 83,637,381,699,544,802,976 & -28,939,583,397,335,447,760 & 7,744,654,310,169,576,800 & -1,634,980,697,246,583,456 & 276,019,109,275,035,346 & -37,600,535,086,859,745 & 4,154,823,851,430,525 & -373,100,999,802,531 & 27,188,611,869,881 & -1,599,718,388,730 & 75,289,668,850 & -2,792,167,686 & 79,721,796 & -1,689,765 & 25,025 & -231 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \hline 23 & 0 & 1,124,000,727,777,607,680,000 & -4,148,476,779,335,454,720,000 & 6,756,146,673,770,930,688,000 & -6,548,684,852,703,068,697,600 & 4,280,722,865,357,147,142,912 & -2,021,687,376,910,682,741,568 & 720,308,216,440,924,653,696 & -199,321,978,221,066,137,360 & 43,714,229,649,594,412,832 & -7,707,401,101,297,361,068 & 1,103,230,881,185,949,736 & -129,006,659,818,331,295 & 12,363,045,847,086,207 & -971,250,460,939,913 & 62,382,416,421,941 & -3,256,091,103,430 & 136,717,357,942 & -4,546,047,198 & 116,896,626 & -2,240,315 & 30,107 & -253 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \hline 24 & 0 & -25,852,016,738,884,976,640,000 & 96,538,966,652,493,066,240,000 & -159,539,850,276,066,860,544,000 & 157,375,898,285,941,510,732,800 & -105,005,310,755,917,452,984,576 & 50,779,532,534,302,850,198,976 & -18,588,776,355,051,949,776,576 & 5,304,713,715,525,445,812,976 & -1,204,749,260,161,737,632,496 & 220,984,454,979,433,717,396 & -33,081,711,368,574,204,996 & 4,070,384,057,007,569,521 & -413,356,714,301,314,056 & 34,701,806,448,704,206 & -2,406,046,038,644,556 & 137,272,511,800,831 & -6,400,590,336,096 & 241,276,443,496 & -7,234,669,596 & 168,423,871 & -2,932,776 & 35,926 & -276 & 1 & 0 & 0 & 0 & 0 & 0 & 0\\ \hline 25 & 0 & 620,448,401,733,239,439,360,000 & -2,342,787,216,398,718,566,400,000 & 3,925,495,373,278,097,719,296,000 & -3,936,561,409,138,663,118,131,200 & 2,677,503,356,427,960,382,362,624 & -1,323,714,091,579,185,857,760,000 & 496,910,165,055,549,644,836,800 & -145,901,905,527,662,649,288,000 & 34,218,695,959,407,148,992,880 & -6,508,376,179,668,146,850,000 & 1,014,945,527,825,214,637,300 & -130,770,928,736,755,873,500 & 13,990,945,200,239,106,865 & -1,246,200,069,070,215,000 & 92,446,911,376,173,550 & -5,700,586,321,864,500 & 290,886,679,867,135 & -12,191,224,980,000 & 414,908,513,800 & -11,276,842,500 & 238,810,495 & -3,795,000 & 42,550 & -300 & 1 & 0 & 0 & 0 & 0 & 0\\ \hline 26 & 0 & -15,511,210,043,330,985,984,000,000 & 59,190,128,811,701,203,599,360,000 & -100,480,171,548,351,161,548,800,000 & 102,339,530,601,744,675,672,576,000 & -70,874,145,319,837,672,677,196,800 & 35,770,355,645,907,606,826,362,624 & -13,746,468,217,967,926,978,680,000 & 4,144,457,803,247,115,877,036,800 & -1,001,369,304,512,841,374,110,000 & 196,928,100,451,110,820,242,880 & -31,882,014,375,298,512,782,500 & 4,284,218,746,244,111,474,800 & -480,544,558,742,733,545,125 & 45,145,946,926,994,481,865 & -3,557,372,853,474,553,750 & 234,961,569,422,786,050 & -12,972,753,318,542,875 & 595,667,304,367,135 & -22,563,937,825,000 & 696,829,576,300 & -17,247,104,875 & 333,685,495 & -4,858,750 & 50,050 & -325 & 1 & 0 & 0 & 0 & 0\\ \hline 27 & 0 & 403,291,461,126,605,635,584,000,000 & -1,554,454,559,147,562,279,567,360,000 & 2,671,674,589,068,831,403,868,160,000 & -2,761,307,967,193,712,729,035,776,000 & 1,945,067,308,917,524,165,279,692,800 & -1,000,903,392,113,435,450,162,625,024 & 393,178,529,313,073,708,272,042,624 & -121,502,371,102,392,939,781,636,800 & 30,180,059,720,580,991,603,896,800 & -6,121,499,916,241,722,700,424,880 & 1,025,860,474,208,872,152,587,880 & -143,271,701,777,645,411,127,300 & 16,778,377,273,555,183,648,050 & -1,654,339,178,844,590,073,615 & 137,637,641,117,332,879,365 & -9,666,373,658,466,991,050 & 572,253,155,704,900,800 & -28,460,103,232,088,385 & 1,182,329,687,817,135 & -40,681,506,808,800 & 1,145,254,303,050 & -25,922,927,745 & 460,012,995 & -6,160,050 & 58,500 & -351 & 1 & 0 & 0 & 0\\ \hline 28 & 0 & -10,888,869,450,418,352,160,768,000,000 & 42,373,564,558,110,787,183,902,720,000 & -73,689,668,464,006,010,184,007,680,000 & 77,226,989,703,299,075,087,834,112,000 & -55,278,125,307,966,865,191,587,481,600 & 28,969,458,895,980,281,319,670,568,448 & -11,616,723,683,566,425,573,507,775,872 & 3,673,742,549,077,683,082,376,236,224 & -936,363,983,558,079,713,086,850,400 & 195,460,557,459,107,504,515,368,560 & -33,819,732,719,881,270,820,297,640 & 4,894,196,422,205,298,253,024,980 & -596,287,888,163,635,369,624,650 & 61,445,535,102,359,115,635,655 & -5,370,555,489,012,577,816,470 & 398,629,729,895,941,637,715 & -25,117,208,862,499,312,650 & 1,340,675,942,971,287,195 & -60,383,004,803,151,030 & 2,280,730,371,654,735 & -71,603,372,991,150 & 1,845,173,352,165 & -38,343,278,610 & 626,334,345 & -7,739,550 & 67,977 & -378 & 1 & 0 & 0\\ \hline 29 & 0 & 304,888,344,611,713,860,501,504,000,000 & -1,197,348,677,077,520,393,310,044,160,000 & 2,105,684,281,550,279,072,336,117,760,000 & -2,236,045,380,156,380,112,643,362,816,000 & 1,625,014,498,326,371,300,452,283,596,800 & -866,422,974,395,414,742,142,363,398,144 & 354,237,722,035,840,197,377,888,292,864 & -114,481,515,057,741,551,880,042,390,144 & 29,891,934,088,703,915,048,808,047,424 & -6,409,259,592,413,089,839,517,170,080 & 1,142,413,073,615,783,087,483,702,480 & -170,857,232,541,629,621,904,997,080 & 21,590,257,290,787,088,602,515,180 & -2,316,762,871,029,690,607,422,990 & 211,821,088,794,711,294,496,815 & -16,532,187,926,098,943,672,490 & 1,101,911,578,045,922,391,915 & -62,656,135,265,695,354,110 & 3,031,400,077,459,516,035 & -124,243,455,209,483,610 & 4,285,624,815,406,935 & -123,268,226,851,770 & 2,918,785,153,245 & -55,880,640,270 & 843,041,745 & -9,642,906 & 78,561 & -406 & 1 & 0\\ \hline 30 & 0 & -8,841,761,993,739,701,954,543,616,000,000 & 35,027,999,979,859,805,266,492,784,640,000 & -62,262,192,842,035,613,491,057,459,200,000 & 66,951,000,306,085,302,338,993,639,424,000 & -49,361,465,831,621,147,825,759,587,123,200 & 26,751,280,755,793,398,822,580,822,142,976 & -11,139,316,913,434,780,466,101,123,891,200 & 3,674,201,658,710,345,201,899,117,607,040 & -981,347,603,630,155,088,295,475,765,440 & 215,760,462,268,683,520,394,805,979,744 & -39,539,238,727,270,799,376,544,542,000 & 6,097,272,817,323,042,122,728,617,800 & -796,974,693,974,455,191,377,937,300 & 88,776,380,550,648,116,217,781,890 & -8,459,574,446,076,318,147,830,625 & 691,254,538,651,580,660,999,025 & -48,487,623,689,430,693,038,025 & 2,918,939,500,751,087,661,105 & -150,566,737,512,021,319,125 & 6,634,460,278,534,540,725 & -248,526,574,856,284,725 & 7,860,403,394,108,265 & -207,912,996,295,875 & 4,539,323,721,075 & -80,328,850,875 & 1,122,686,019 & -11,921,175 & 90,335 & -435 & 1 \\\hline \end{array} \] 第2種スターリング数\(S_{2}\left(n,k\right)\)
\[ \begin{array}{|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|} \hline n\backslash k & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 & 13 & 14 & 15 & 16 & 17 & 18 & 19 & 20 & 21 & 22 & 23 & 24 & 25 & 26 & 27 & 28 & 29 & 30\\ \hline 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \hline 1 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \hline 2 & 0 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \hline 3 & 0 & 1 & 3 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \hline 4 & 0 & 1 & 7 & 6 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \hline 5 & 0 & 1 & 15 & 25 & 10 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \hline 6 & 0 & 1 & 31 & 90 & 65 & 15 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \hline 7 & 0 & 1 & 63 & 301 & 350 & 140 & 21 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \hline 8 & 0 & 1 & 127 & 966 & 1,701 & 1,050 & 266 & 28 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \hline 9 & 0 & 1 & 255 & 3,025 & 7,770 & 6,951 & 2,646 & 462 & 36 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \hline 10 & 0 & 1 & 511 & 9,330 & 34,105 & 42,525 & 22,827 & 5,880 & 750 & 45 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \hline 11 & 0 & 1 & 1,023 & 28,501 & 145,750 & 246,730 & 179,487 & 63,987 & 11,880 & 1,155 & 55 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \hline 12 & 0 & 1 & 2,047 & 86,526 & 611,501 & 1,379,400 & 1,323,652 & 627,396 & 159,027 & 22,275 & 1,705 & 66 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \hline 13 & 0 & 1 & 4,095 & 261,625 & 2,532,530 & 7,508,501 & 9,321,312 & 5,715,424 & 1,899,612 & 359,502 & 39,325 & 2,431 & 78 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \hline 14 & 0 & 1 & 8,191 & 788,970 & 10,391,745 & 40,075,035 & 63,436,373 & 49,329,280 & 20,912,320 & 5,135,130 & 752,752 & 66,066 & 3,367 & 91 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \hline 15 & 0 & 1 & 16,383 & 2,375,101 & 42,355,950 & 210,766,920 & 420,693,273 & 408,741,333 & 216,627,840 & 67,128,490 & 12,662,650 & 1,479,478 & 106,470 & 4,550 & 105 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \hline 16 & 0 & 1 & 32,767 & 7,141,686 & 171,798,901 & 1,096,190,550 & 2,734,926,558 & 3,281,882,604 & 2,141,764,053 & 820,784,250 & 193,754,990 & 28,936,908 & 2,757,118 & 165,620 & 6,020 & 120 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \hline 17 & 0 & 1 & 65,535 & 21,457,825 & 694,337,290 & 5,652,751,651 & 17,505,749,898 & 25,708,104,786 & 20,415,995,028 & 9,528,822,303 & 2,758,334,150 & 512,060,978 & 62,022,324 & 4,910,178 & 249,900 & 7,820 & 136 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \hline 18 & 0 & 1 & 131,071 & 64,439,010 & 2,798,806,985 & 28,958,095,545 & 110,687,251,039 & 197,462,483,400 & 189,036,065,010 & 106,175,395,755 & 37,112,163,803 & 8,391,004,908 & 1,256,328,866 & 125,854,638 & 8,408,778 & 367,200 & 9,996 & 153 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \hline 19 & 0 & 1 & 262,143 & 193,448,101 & 11,259,666,950 & 147,589,284,710 & 693,081,601,779 & 1,492,924,634,839 & 1,709,751,003,480 & 1,144,614,626,805 & 477,297,033,785 & 129,413,217,791 & 23,466,951,300 & 2,892,439,160 & 243,577,530 & 13,916,778 & 527,136 & 12,597 & 171 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \hline 20 & 0 & 1 & 524,287 & 580,606,446 & 45,232,115,901 & 749,206,090,500 & 4,306,078,895,384 & 11,143,554,045,652 & 15,170,932,662,679 & 12,011,282,644,725 & 5,917,584,964,655 & 1,900,842,429,486 & 411,016,633,391 & 61,068,660,380 & 6,302,524,580 & 452,329,200 & 22,350,954 & 741,285 & 15,675 & 190 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \hline 21 & 0 & 1 & 1,048,575 & 1,742,343,625 & 181,509,070,050 & 3,791,262,568,401 & 26,585,679,462,804 & 82,310,957,214,948 & 132,511,015,347,084 & 123,272,476,465,204 & 71,187,132,291,275 & 26,826,851,689,001 & 6,833,042,030,178 & 1,204,909,218,331 & 149,304,004,500 & 13,087,462,580 & 809,944,464 & 34,952,799 & 1,023,435 & 19,285 & 210 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \hline 22 & 0 & 1 & 2,097,151 & 5,228,079,450 & 727,778,623,825 & 19,137,821,912,055 & 163,305,339,345,225 & 602,762,379,967,440 & 1,142,399,079,991,620 & 1,241,963,303,533,920 & 835,143,799,377,954 & 366,282,500,870,286 & 108,823,356,051,137 & 22,496,861,868,481 & 3,295,165,281,331 & 345,615,943,200 & 26,046,574,004 & 1,404,142,047 & 53,374,629 & 1,389,850 & 23,485 & 231 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \hline 23 & 0 & 1 & 4,194,303 & 15,686,335,501 & 2,916,342,574,750 & 96,416,888,184,100 & 998,969,857,983,405 & 4,382,641,999,117,305 & 9,741,955,019,900,400 & 12,320,068,811,796,900 & 9,593,401,297,313,460 & 4,864,251,308,951,100 & 1,672,162,773,483,930 & 401,282,560,341,390 & 68,629,175,807,115 & 8,479,404,429,331 & 762,361,127,264 & 49,916,988,803 & 2,364,885,369 & 79,781,779 & 1,859,550 & 28,336 & 253 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \hline 24 & 0 & 1 & 8,388,607 & 47,063,200,806 & 11,681,056,634,501 & 485,000,783,495,250 & 6,090,236,036,084,530 & 31,677,463,851,804,540 & 82,318,282,158,320,505 & 120,622,574,326,072,500 & 108,254,081,784,931,500 & 63,100,165,695,775,560 & 24,930,204,590,758,260 & 6,888,836,057,922,000 & 1,362,091,021,641,000 & 195,820,242,247,080 & 20,677,182,465,555 & 1,610,949,936,915 & 92,484,925,445 & 3,880,739,170 & 116,972,779 & 2,454,606 & 33,902 & 276 & 1 & 0 & 0 & 0 & 0 & 0 & 0\\ \hline 25 & 0 & 1 & 16,777,215 & 141,197,991,025 & 46,771,289,738,810 & 2,436,684,974,110,751 & 37,026,417,000,002,430 & 227,832,482,998,716,310 & 690,223,721,118,368,580 & 1,167,921,451,092,973,005 & 1,203,163,392,175,387,500 & 802,355,904,438,462,660 & 362,262,620,784,874,680 & 114,485,073,343,744,260 & 25,958,110,360,896,000 & 4,299,394,655,347,200 & 526,655,161,695,960 & 48,063,331,393,110 & 3,275,678,594,925 & 166,218,969,675 & 6,220,194,750 & 168,519,505 & 3,200,450 & 40,250 & 300 & 1 & 0 & 0 & 0 & 0 & 0\\ \hline 26 & 0 & 1 & 33,554,431 & 423,610,750,290 & 187,226,356,946,265 & 12,230,196,160,292,565 & 224,595,186,974,125,331 & 1,631,853,797,991,016,600 & 5,749,622,251,945,664,950 & 11,201,516,780,955,125,625 & 13,199,555,372,846,848,005 & 10,029,078,340,998,476,760 & 5,149,507,353,856,958,820 & 1,850,568,574,253,550,060 & 477,898,618,396,288,260 & 90,449,030,191,104,000 & 12,725,877,242,482,560 & 1,343,731,795,378,830 & 107,025,546,101,760 & 6,433,839,018,750 & 290,622,864,675 & 9,759,104,355 & 238,929,405 & 4,126,200 & 47,450 & 325 & 1 & 0 & 0 & 0 & 0\\ \hline 27 & 0 & 1 & 67,108,863 & 1,270,865,805,301 & 749,329,038,535,350 & 61,338,207,158,409,090 & 1,359,801,318,005,044,551 & 11,647,571,772,911,241,531 & 47,628,831,813,556,336,200 & 106,563,273,280,541,795,575 & 143,197,070,509,423,605,675 & 123,519,417,123,830,092,365 & 71,823,166,587,281,982,600 & 29,206,898,819,153,109,600 & 8,541,149,231,801,585,700 & 1,834,634,071,262,848,260 & 294,063,066,070,824,960 & 35,569,317,763,922,670 & 3,270,191,625,210,510 & 229,268,487,458,010 & 12,246,296,312,250 & 495,564,056,130 & 15,015,551,265 & 333,832,005 & 5,265,000 & 55,575 & 351 & 1 & 0 & 0 & 0\\ \hline 28 & 0 & 1 & 134,217,727 & 3,812,664,524,766 & 2,998,587,019,946,701 & 307,440,364,830,580,800 & 8,220,146,115,188,676,396 & 82,892,803,728,383,735,268 & 392,678,226,281,361,931,131 & 1,006,698,291,338,432,496,375 & 1,538,533,978,374,777,852,325 & 1,501,910,658,871,554,621,690 & 985,397,416,171,213,883,565 & 451,512,851,236,272,407,400 & 148,782,988,064,375,309,400 & 36,060,660,300,744,309,600 & 6,539,643,128,396,047,620 & 898,741,468,057,510,350 & 94,432,767,017,711,850 & 7,626,292,886,912,700 & 474,194,413,703,010 & 22,653,141,490,980 & 825,906,183,960 & 22,693,687,380 & 460,192,005 & 6,654,375 & 64,701 & 378 & 1 & 0 & 0\\ \hline 29 & 0 & 1 & 268,435,455 & 11,438,127,792,025 & 11,998,160,744,311,570 & 1,540,200,411,172,850,701 & 49,628,317,055,962,639,176 & 588,469,772,213,874,823,272 & 3,224,318,613,979,279,184,316 & 9,452,962,848,327,254,398,506 & 16,392,038,075,086,211,019,625 & 18,059,551,225,961,878,690,915 & 13,326,679,652,926,121,224,470 & 6,855,064,482,242,755,179,765 & 2,534,474,684,137,526,739,000 & 689,692,892,575,539,953,400 & 140,694,950,355,081,071,520 & 21,818,248,085,373,723,570 & 2,598,531,274,376,323,650 & 239,332,331,869,053,150 & 17,110,181,160,972,900 & 949,910,385,013,590 & 40,823,077,538,100 & 1,347,860,993,700 & 33,738,295,500 & 626,551,380 & 8,336,601 & 74,907 & 406 & 1 & 0\\ \hline 30 & 0 & 1 & 536,870,911 & 34,314,651,811,530 & 48,004,081,105,038,305 & 7,713,000,216,608,565,075 & 299,310,102,746,948,685,757 & 4,168,916,722,553,086,402,080 & 26,383,018,684,048,108,297,800 & 88,300,984,248,924,568,770,870 & 173,373,343,599,189,364,594,756 & 215,047,101,560,666,876,619,690 & 177,979,707,061,075,333,384,555 & 102,442,517,922,081,938,561,415 & 42,337,710,060,168,129,525,765 & 12,879,868,072,770,626,040,000 & 2,940,812,098,256,837,097,720 & 511,605,167,806,434,372,210 & 68,591,811,024,147,549,270 & 7,145,845,579,888,333,500 & 581,535,955,088,511,150 & 37,058,299,246,258,290 & 1,848,018,090,851,790 & 71,823,880,393,200 & 2,157,580,085,700 & 49,402,080,000 & 843,303,006 & 10,359,090 & 86,275 & 435 & 1 \\\hline \end{array} \]
\[ \begin{array}{|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|} \hline n\backslash k & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 & 13 & 14 & 15 & 16 & 17 & 18 & 19 & 20 & 21 & 22 & 23 & 24 & 25 & 26 & 27 & 28 & 29 & 30\\ \hline 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \hline 1 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \hline 2 & 0 & -1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \hline 3 & 0 & 2 & -3 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \hline 4 & 0 & -6 & 11 & -6 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \hline 5 & 0 & 24 & -50 & 35 & -10 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \hline 6 & 0 & -120 & 274 & -225 & 85 & -15 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \hline 7 & 0 & 720 & -1,764 & 1,624 & -735 & 175 & -21 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \hline 8 & 0 & -5,040 & 13,068 & -13,132 & 6,769 & -1,960 & 322 & -28 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \hline 9 & 0 & 40,320 & -109,584 & 118,124 & -67,284 & 22,449 & -4,536 & 546 & -36 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \hline 10 & 0 & -362,880 & 1,026,576 & -1,172,700 & 723,680 & -269,325 & 63,273 & -9,450 & 870 & -45 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \hline 11 & 0 & 3,628,800 & -10,628,640 & 12,753,576 & -8,409,500 & 3,416,930 & -902,055 & 157,773 & -18,150 & 1,320 & -55 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \hline 12 & 0 & -39,916,800 & 120,543,840 & -150,917,976 & 105,258,076 & -45,995,730 & 13,339,535 & -2,637,558 & 357,423 & -32,670 & 1,925 & -66 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \hline 13 & 0 & 479,001,600 & -1,486,442,880 & 1,931,559,552 & -1,414,014,888 & 657,206,836 & -206,070,150 & 44,990,231 & -6,926,634 & 749,463 & -55,770 & 2,717 & -78 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \hline 14 & 0 & -6,227,020,800 & 19,802,759,040 & -26,596,717,056 & 20,313,753,096 & -9,957,703,756 & 3,336,118,786 & -790,943,153 & 135,036,473 & -16,669,653 & 1,474,473 & -91,091 & 3,731 & -91 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \hline 15 & 0 & 87,178,291,200 & -283,465,647,360 & 392,156,797,824 & -310,989,260,400 & 159,721,605,680 & -56,663,366,760 & 14,409,322,928 & -2,681,453,775 & 368,411,615 & -37,312,275 & 2,749,747 & -143,325 & 5,005 & -105 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \hline 16 & 0 & -1,307,674,368,000 & 4,339,163,001,600 & -6,165,817,614,720 & 5,056,995,703,824 & -2,706,813,345,600 & 1,009,672,107,080 & -272,803,210,680 & 54,631,129,553 & -8,207,628,000 & 928,095,740 & -78,558,480 & 4,899,622 & -218,400 & 6,580 & -120 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \hline 17 & 0 & 20,922,789,888,000 & -70,734,282,393,600 & 102,992,244,837,120 & -87,077,748,875,904 & 48,366,009,233,424 & -18,861,567,058,880 & 5,374,523,477,960 & -1,146,901,283,528 & 185,953,177,553 & -23,057,159,840 & 2,185,031,420 & -156,952,432 & 8,394,022 & -323,680 & 8,500 & -136 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \hline 18 & 0 & -355,687,428,096,000 & 1,223,405,590,579,200 & -1,821,602,444,624,640 & 1,583,313,975,727,488 & -909,299,905,844,112 & 369,012,649,234,384 & -110,228,466,184,200 & 24,871,845,297,936 & -4,308,105,301,929 & 577,924,894,833 & -60,202,693,980 & 4,853,222,764 & -299,650,806 & 13,896,582 & -468,180 & 10,812 & -153 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \hline 19 & 0 & 6,402,373,705,728,000 & -22,376,988,058,521,600 & 34,012,249,593,822,720 & -30,321,254,007,719,424 & 17,950,712,280,921,504 & -7,551,527,592,063,024 & 2,353,125,040,549,984 & -557,921,681,547,048 & 102,417,740,732,658 & -14,710,753,408,923 & 1,661,573,386,473 & -147,560,703,732 & 10,246,937,272 & -549,789,282 & 22,323,822 & -662,796 & 13,566 & -171 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \hline 20 & 0 & -121,645,100,408,832,000 & 431,565,146,817,638,400 & -668,609,730,341,153,280 & 610,116,075,740,491,776 & -371,384,787,345,228,000 & 161,429,736,530,118,960 & -52,260,903,362,512,720 & 12,953,636,989,943,896 & -2,503,858,755,467,550 & 381,922,055,502,195 & -46,280,647,751,910 & 4,465,226,757,381 & -342,252,511,900 & 20,692,933,630 & -973,941,900 & 34,916,946 & -920,550 & 16,815 & -190 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \hline 21 & 0 & 2,432,902,008,176,640,000 & -8,752,948,036,761,600,000 & 13,803,759,753,640,704,000 & -12,870,931,245,150,988,800 & 8,037,811,822,645,051,776 & -3,599,979,517,947,607,200 & 1,206,647,803,780,373,360 & -311,333,643,161,390,640 & 63,030,812,099,294,896 & -10,142,299,865,511,450 & 1,307,535,010,540,395 & -135,585,182,899,530 & 11,310,276,995,381 & -756,111,184,500 & 40,171,771,630 & -1,672,280,820 & 53,327,946 & -1,256,850 & 20,615 & -210 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \hline 22 & 0 & -51,090,942,171,709,440,000 & 186,244,810,780,170,240,000 & -298,631,902,863,216,384,000 & 284,093,315,901,811,468,800 & -181,664,979,520,697,076,096 & 83,637,381,699,544,802,976 & -28,939,583,397,335,447,760 & 7,744,654,310,169,576,800 & -1,634,980,697,246,583,456 & 276,019,109,275,035,346 & -37,600,535,086,859,745 & 4,154,823,851,430,525 & -373,100,999,802,531 & 27,188,611,869,881 & -1,599,718,388,730 & 75,289,668,850 & -2,792,167,686 & 79,721,796 & -1,689,765 & 25,025 & -231 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \hline 23 & 0 & 1,124,000,727,777,607,680,000 & -4,148,476,779,335,454,720,000 & 6,756,146,673,770,930,688,000 & -6,548,684,852,703,068,697,600 & 4,280,722,865,357,147,142,912 & -2,021,687,376,910,682,741,568 & 720,308,216,440,924,653,696 & -199,321,978,221,066,137,360 & 43,714,229,649,594,412,832 & -7,707,401,101,297,361,068 & 1,103,230,881,185,949,736 & -129,006,659,818,331,295 & 12,363,045,847,086,207 & -971,250,460,939,913 & 62,382,416,421,941 & -3,256,091,103,430 & 136,717,357,942 & -4,546,047,198 & 116,896,626 & -2,240,315 & 30,107 & -253 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \hline 24 & 0 & -25,852,016,738,884,976,640,000 & 96,538,966,652,493,066,240,000 & -159,539,850,276,066,860,544,000 & 157,375,898,285,941,510,732,800 & -105,005,310,755,917,452,984,576 & 50,779,532,534,302,850,198,976 & -18,588,776,355,051,949,776,576 & 5,304,713,715,525,445,812,976 & -1,204,749,260,161,737,632,496 & 220,984,454,979,433,717,396 & -33,081,711,368,574,204,996 & 4,070,384,057,007,569,521 & -413,356,714,301,314,056 & 34,701,806,448,704,206 & -2,406,046,038,644,556 & 137,272,511,800,831 & -6,400,590,336,096 & 241,276,443,496 & -7,234,669,596 & 168,423,871 & -2,932,776 & 35,926 & -276 & 1 & 0 & 0 & 0 & 0 & 0 & 0\\ \hline 25 & 0 & 620,448,401,733,239,439,360,000 & -2,342,787,216,398,718,566,400,000 & 3,925,495,373,278,097,719,296,000 & -3,936,561,409,138,663,118,131,200 & 2,677,503,356,427,960,382,362,624 & -1,323,714,091,579,185,857,760,000 & 496,910,165,055,549,644,836,800 & -145,901,905,527,662,649,288,000 & 34,218,695,959,407,148,992,880 & -6,508,376,179,668,146,850,000 & 1,014,945,527,825,214,637,300 & -130,770,928,736,755,873,500 & 13,990,945,200,239,106,865 & -1,246,200,069,070,215,000 & 92,446,911,376,173,550 & -5,700,586,321,864,500 & 290,886,679,867,135 & -12,191,224,980,000 & 414,908,513,800 & -11,276,842,500 & 238,810,495 & -3,795,000 & 42,550 & -300 & 1 & 0 & 0 & 0 & 0 & 0\\ \hline 26 & 0 & -15,511,210,043,330,985,984,000,000 & 59,190,128,811,701,203,599,360,000 & -100,480,171,548,351,161,548,800,000 & 102,339,530,601,744,675,672,576,000 & -70,874,145,319,837,672,677,196,800 & 35,770,355,645,907,606,826,362,624 & -13,746,468,217,967,926,978,680,000 & 4,144,457,803,247,115,877,036,800 & -1,001,369,304,512,841,374,110,000 & 196,928,100,451,110,820,242,880 & -31,882,014,375,298,512,782,500 & 4,284,218,746,244,111,474,800 & -480,544,558,742,733,545,125 & 45,145,946,926,994,481,865 & -3,557,372,853,474,553,750 & 234,961,569,422,786,050 & -12,972,753,318,542,875 & 595,667,304,367,135 & -22,563,937,825,000 & 696,829,576,300 & -17,247,104,875 & 333,685,495 & -4,858,750 & 50,050 & -325 & 1 & 0 & 0 & 0 & 0\\ \hline 27 & 0 & 403,291,461,126,605,635,584,000,000 & -1,554,454,559,147,562,279,567,360,000 & 2,671,674,589,068,831,403,868,160,000 & -2,761,307,967,193,712,729,035,776,000 & 1,945,067,308,917,524,165,279,692,800 & -1,000,903,392,113,435,450,162,625,024 & 393,178,529,313,073,708,272,042,624 & -121,502,371,102,392,939,781,636,800 & 30,180,059,720,580,991,603,896,800 & -6,121,499,916,241,722,700,424,880 & 1,025,860,474,208,872,152,587,880 & -143,271,701,777,645,411,127,300 & 16,778,377,273,555,183,648,050 & -1,654,339,178,844,590,073,615 & 137,637,641,117,332,879,365 & -9,666,373,658,466,991,050 & 572,253,155,704,900,800 & -28,460,103,232,088,385 & 1,182,329,687,817,135 & -40,681,506,808,800 & 1,145,254,303,050 & -25,922,927,745 & 460,012,995 & -6,160,050 & 58,500 & -351 & 1 & 0 & 0 & 0\\ \hline 28 & 0 & -10,888,869,450,418,352,160,768,000,000 & 42,373,564,558,110,787,183,902,720,000 & -73,689,668,464,006,010,184,007,680,000 & 77,226,989,703,299,075,087,834,112,000 & -55,278,125,307,966,865,191,587,481,600 & 28,969,458,895,980,281,319,670,568,448 & -11,616,723,683,566,425,573,507,775,872 & 3,673,742,549,077,683,082,376,236,224 & -936,363,983,558,079,713,086,850,400 & 195,460,557,459,107,504,515,368,560 & -33,819,732,719,881,270,820,297,640 & 4,894,196,422,205,298,253,024,980 & -596,287,888,163,635,369,624,650 & 61,445,535,102,359,115,635,655 & -5,370,555,489,012,577,816,470 & 398,629,729,895,941,637,715 & -25,117,208,862,499,312,650 & 1,340,675,942,971,287,195 & -60,383,004,803,151,030 & 2,280,730,371,654,735 & -71,603,372,991,150 & 1,845,173,352,165 & -38,343,278,610 & 626,334,345 & -7,739,550 & 67,977 & -378 & 1 & 0 & 0\\ \hline 29 & 0 & 304,888,344,611,713,860,501,504,000,000 & -1,197,348,677,077,520,393,310,044,160,000 & 2,105,684,281,550,279,072,336,117,760,000 & -2,236,045,380,156,380,112,643,362,816,000 & 1,625,014,498,326,371,300,452,283,596,800 & -866,422,974,395,414,742,142,363,398,144 & 354,237,722,035,840,197,377,888,292,864 & -114,481,515,057,741,551,880,042,390,144 & 29,891,934,088,703,915,048,808,047,424 & -6,409,259,592,413,089,839,517,170,080 & 1,142,413,073,615,783,087,483,702,480 & -170,857,232,541,629,621,904,997,080 & 21,590,257,290,787,088,602,515,180 & -2,316,762,871,029,690,607,422,990 & 211,821,088,794,711,294,496,815 & -16,532,187,926,098,943,672,490 & 1,101,911,578,045,922,391,915 & -62,656,135,265,695,354,110 & 3,031,400,077,459,516,035 & -124,243,455,209,483,610 & 4,285,624,815,406,935 & -123,268,226,851,770 & 2,918,785,153,245 & -55,880,640,270 & 843,041,745 & -9,642,906 & 78,561 & -406 & 1 & 0\\ \hline 30 & 0 & -8,841,761,993,739,701,954,543,616,000,000 & 35,027,999,979,859,805,266,492,784,640,000 & -62,262,192,842,035,613,491,057,459,200,000 & 66,951,000,306,085,302,338,993,639,424,000 & -49,361,465,831,621,147,825,759,587,123,200 & 26,751,280,755,793,398,822,580,822,142,976 & -11,139,316,913,434,780,466,101,123,891,200 & 3,674,201,658,710,345,201,899,117,607,040 & -981,347,603,630,155,088,295,475,765,440 & 215,760,462,268,683,520,394,805,979,744 & -39,539,238,727,270,799,376,544,542,000 & 6,097,272,817,323,042,122,728,617,800 & -796,974,693,974,455,191,377,937,300 & 88,776,380,550,648,116,217,781,890 & -8,459,574,446,076,318,147,830,625 & 691,254,538,651,580,660,999,025 & -48,487,623,689,430,693,038,025 & 2,918,939,500,751,087,661,105 & -150,566,737,512,021,319,125 & 6,634,460,278,534,540,725 & -248,526,574,856,284,725 & 7,860,403,394,108,265 & -207,912,996,295,875 & 4,539,323,721,075 & -80,328,850,875 & 1,122,686,019 & -11,921,175 & 90,335 & -435 & 1 \\\hline \end{array} \] 第2種スターリング数\(S_{2}\left(n,k\right)\)
\[ \begin{array}{|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|} \hline n\backslash k & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 & 13 & 14 & 15 & 16 & 17 & 18 & 19 & 20 & 21 & 22 & 23 & 24 & 25 & 26 & 27 & 28 & 29 & 30\\ \hline 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \hline 1 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \hline 2 & 0 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \hline 3 & 0 & 1 & 3 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \hline 4 & 0 & 1 & 7 & 6 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \hline 5 & 0 & 1 & 15 & 25 & 10 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \hline 6 & 0 & 1 & 31 & 90 & 65 & 15 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \hline 7 & 0 & 1 & 63 & 301 & 350 & 140 & 21 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \hline 8 & 0 & 1 & 127 & 966 & 1,701 & 1,050 & 266 & 28 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \hline 9 & 0 & 1 & 255 & 3,025 & 7,770 & 6,951 & 2,646 & 462 & 36 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \hline 10 & 0 & 1 & 511 & 9,330 & 34,105 & 42,525 & 22,827 & 5,880 & 750 & 45 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \hline 11 & 0 & 1 & 1,023 & 28,501 & 145,750 & 246,730 & 179,487 & 63,987 & 11,880 & 1,155 & 55 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \hline 12 & 0 & 1 & 2,047 & 86,526 & 611,501 & 1,379,400 & 1,323,652 & 627,396 & 159,027 & 22,275 & 1,705 & 66 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \hline 13 & 0 & 1 & 4,095 & 261,625 & 2,532,530 & 7,508,501 & 9,321,312 & 5,715,424 & 1,899,612 & 359,502 & 39,325 & 2,431 & 78 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \hline 14 & 0 & 1 & 8,191 & 788,970 & 10,391,745 & 40,075,035 & 63,436,373 & 49,329,280 & 20,912,320 & 5,135,130 & 752,752 & 66,066 & 3,367 & 91 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \hline 15 & 0 & 1 & 16,383 & 2,375,101 & 42,355,950 & 210,766,920 & 420,693,273 & 408,741,333 & 216,627,840 & 67,128,490 & 12,662,650 & 1,479,478 & 106,470 & 4,550 & 105 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \hline 16 & 0 & 1 & 32,767 & 7,141,686 & 171,798,901 & 1,096,190,550 & 2,734,926,558 & 3,281,882,604 & 2,141,764,053 & 820,784,250 & 193,754,990 & 28,936,908 & 2,757,118 & 165,620 & 6,020 & 120 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \hline 17 & 0 & 1 & 65,535 & 21,457,825 & 694,337,290 & 5,652,751,651 & 17,505,749,898 & 25,708,104,786 & 20,415,995,028 & 9,528,822,303 & 2,758,334,150 & 512,060,978 & 62,022,324 & 4,910,178 & 249,900 & 7,820 & 136 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \hline 18 & 0 & 1 & 131,071 & 64,439,010 & 2,798,806,985 & 28,958,095,545 & 110,687,251,039 & 197,462,483,400 & 189,036,065,010 & 106,175,395,755 & 37,112,163,803 & 8,391,004,908 & 1,256,328,866 & 125,854,638 & 8,408,778 & 367,200 & 9,996 & 153 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \hline 19 & 0 & 1 & 262,143 & 193,448,101 & 11,259,666,950 & 147,589,284,710 & 693,081,601,779 & 1,492,924,634,839 & 1,709,751,003,480 & 1,144,614,626,805 & 477,297,033,785 & 129,413,217,791 & 23,466,951,300 & 2,892,439,160 & 243,577,530 & 13,916,778 & 527,136 & 12,597 & 171 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \hline 20 & 0 & 1 & 524,287 & 580,606,446 & 45,232,115,901 & 749,206,090,500 & 4,306,078,895,384 & 11,143,554,045,652 & 15,170,932,662,679 & 12,011,282,644,725 & 5,917,584,964,655 & 1,900,842,429,486 & 411,016,633,391 & 61,068,660,380 & 6,302,524,580 & 452,329,200 & 22,350,954 & 741,285 & 15,675 & 190 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \hline 21 & 0 & 1 & 1,048,575 & 1,742,343,625 & 181,509,070,050 & 3,791,262,568,401 & 26,585,679,462,804 & 82,310,957,214,948 & 132,511,015,347,084 & 123,272,476,465,204 & 71,187,132,291,275 & 26,826,851,689,001 & 6,833,042,030,178 & 1,204,909,218,331 & 149,304,004,500 & 13,087,462,580 & 809,944,464 & 34,952,799 & 1,023,435 & 19,285 & 210 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \hline 22 & 0 & 1 & 2,097,151 & 5,228,079,450 & 727,778,623,825 & 19,137,821,912,055 & 163,305,339,345,225 & 602,762,379,967,440 & 1,142,399,079,991,620 & 1,241,963,303,533,920 & 835,143,799,377,954 & 366,282,500,870,286 & 108,823,356,051,137 & 22,496,861,868,481 & 3,295,165,281,331 & 345,615,943,200 & 26,046,574,004 & 1,404,142,047 & 53,374,629 & 1,389,850 & 23,485 & 231 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \hline 23 & 0 & 1 & 4,194,303 & 15,686,335,501 & 2,916,342,574,750 & 96,416,888,184,100 & 998,969,857,983,405 & 4,382,641,999,117,305 & 9,741,955,019,900,400 & 12,320,068,811,796,900 & 9,593,401,297,313,460 & 4,864,251,308,951,100 & 1,672,162,773,483,930 & 401,282,560,341,390 & 68,629,175,807,115 & 8,479,404,429,331 & 762,361,127,264 & 49,916,988,803 & 2,364,885,369 & 79,781,779 & 1,859,550 & 28,336 & 253 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \hline 24 & 0 & 1 & 8,388,607 & 47,063,200,806 & 11,681,056,634,501 & 485,000,783,495,250 & 6,090,236,036,084,530 & 31,677,463,851,804,540 & 82,318,282,158,320,505 & 120,622,574,326,072,500 & 108,254,081,784,931,500 & 63,100,165,695,775,560 & 24,930,204,590,758,260 & 6,888,836,057,922,000 & 1,362,091,021,641,000 & 195,820,242,247,080 & 20,677,182,465,555 & 1,610,949,936,915 & 92,484,925,445 & 3,880,739,170 & 116,972,779 & 2,454,606 & 33,902 & 276 & 1 & 0 & 0 & 0 & 0 & 0 & 0\\ \hline 25 & 0 & 1 & 16,777,215 & 141,197,991,025 & 46,771,289,738,810 & 2,436,684,974,110,751 & 37,026,417,000,002,430 & 227,832,482,998,716,310 & 690,223,721,118,368,580 & 1,167,921,451,092,973,005 & 1,203,163,392,175,387,500 & 802,355,904,438,462,660 & 362,262,620,784,874,680 & 114,485,073,343,744,260 & 25,958,110,360,896,000 & 4,299,394,655,347,200 & 526,655,161,695,960 & 48,063,331,393,110 & 3,275,678,594,925 & 166,218,969,675 & 6,220,194,750 & 168,519,505 & 3,200,450 & 40,250 & 300 & 1 & 0 & 0 & 0 & 0 & 0\\ \hline 26 & 0 & 1 & 33,554,431 & 423,610,750,290 & 187,226,356,946,265 & 12,230,196,160,292,565 & 224,595,186,974,125,331 & 1,631,853,797,991,016,600 & 5,749,622,251,945,664,950 & 11,201,516,780,955,125,625 & 13,199,555,372,846,848,005 & 10,029,078,340,998,476,760 & 5,149,507,353,856,958,820 & 1,850,568,574,253,550,060 & 477,898,618,396,288,260 & 90,449,030,191,104,000 & 12,725,877,242,482,560 & 1,343,731,795,378,830 & 107,025,546,101,760 & 6,433,839,018,750 & 290,622,864,675 & 9,759,104,355 & 238,929,405 & 4,126,200 & 47,450 & 325 & 1 & 0 & 0 & 0 & 0\\ \hline 27 & 0 & 1 & 67,108,863 & 1,270,865,805,301 & 749,329,038,535,350 & 61,338,207,158,409,090 & 1,359,801,318,005,044,551 & 11,647,571,772,911,241,531 & 47,628,831,813,556,336,200 & 106,563,273,280,541,795,575 & 143,197,070,509,423,605,675 & 123,519,417,123,830,092,365 & 71,823,166,587,281,982,600 & 29,206,898,819,153,109,600 & 8,541,149,231,801,585,700 & 1,834,634,071,262,848,260 & 294,063,066,070,824,960 & 35,569,317,763,922,670 & 3,270,191,625,210,510 & 229,268,487,458,010 & 12,246,296,312,250 & 495,564,056,130 & 15,015,551,265 & 333,832,005 & 5,265,000 & 55,575 & 351 & 1 & 0 & 0 & 0\\ \hline 28 & 0 & 1 & 134,217,727 & 3,812,664,524,766 & 2,998,587,019,946,701 & 307,440,364,830,580,800 & 8,220,146,115,188,676,396 & 82,892,803,728,383,735,268 & 392,678,226,281,361,931,131 & 1,006,698,291,338,432,496,375 & 1,538,533,978,374,777,852,325 & 1,501,910,658,871,554,621,690 & 985,397,416,171,213,883,565 & 451,512,851,236,272,407,400 & 148,782,988,064,375,309,400 & 36,060,660,300,744,309,600 & 6,539,643,128,396,047,620 & 898,741,468,057,510,350 & 94,432,767,017,711,850 & 7,626,292,886,912,700 & 474,194,413,703,010 & 22,653,141,490,980 & 825,906,183,960 & 22,693,687,380 & 460,192,005 & 6,654,375 & 64,701 & 378 & 1 & 0 & 0\\ \hline 29 & 0 & 1 & 268,435,455 & 11,438,127,792,025 & 11,998,160,744,311,570 & 1,540,200,411,172,850,701 & 49,628,317,055,962,639,176 & 588,469,772,213,874,823,272 & 3,224,318,613,979,279,184,316 & 9,452,962,848,327,254,398,506 & 16,392,038,075,086,211,019,625 & 18,059,551,225,961,878,690,915 & 13,326,679,652,926,121,224,470 & 6,855,064,482,242,755,179,765 & 2,534,474,684,137,526,739,000 & 689,692,892,575,539,953,400 & 140,694,950,355,081,071,520 & 21,818,248,085,373,723,570 & 2,598,531,274,376,323,650 & 239,332,331,869,053,150 & 17,110,181,160,972,900 & 949,910,385,013,590 & 40,823,077,538,100 & 1,347,860,993,700 & 33,738,295,500 & 626,551,380 & 8,336,601 & 74,907 & 406 & 1 & 0\\ \hline 30 & 0 & 1 & 536,870,911 & 34,314,651,811,530 & 48,004,081,105,038,305 & 7,713,000,216,608,565,075 & 299,310,102,746,948,685,757 & 4,168,916,722,553,086,402,080 & 26,383,018,684,048,108,297,800 & 88,300,984,248,924,568,770,870 & 173,373,343,599,189,364,594,756 & 215,047,101,560,666,876,619,690 & 177,979,707,061,075,333,384,555 & 102,442,517,922,081,938,561,415 & 42,337,710,060,168,129,525,765 & 12,879,868,072,770,626,040,000 & 2,940,812,098,256,837,097,720 & 511,605,167,806,434,372,210 & 68,591,811,024,147,549,270 & 7,145,845,579,888,333,500 & 581,535,955,088,511,150 & 37,058,299,246,258,290 & 1,848,018,090,851,790 & 71,823,880,393,200 & 2,157,580,085,700 & 49,402,080,000 & 843,303,006 & 10,359,090 & 86,275 & 435 & 1 \\\hline \end{array} \]
第1種スターリング数は
\[ Q\left(x,n\right)=\sum_{k=0}^{n}S_{1}'\left(n,k\right)x^{k} \] で定義する場合もある。
この定義と、
\[ P\left(x,n\right)=\sum_{k=0}^{n}S_{1}\left(n,k\right)x^{k} \] で定義した場合と比べる。
これらの第1種スターリング数同士の関係は、
\begin{align*} \sum_{k=0}^{n}S_{1}'\left(n,k\right)x^{k} & =Q\left(x,n\right)\\ & =\left(-1\right)^{n}P\left(-x,n\right)\\ & =\left(-1\right)^{n}\sum_{k=0}^{n}S_{1}\left(n,k\right)\left(-x\right)^{k}\\ & =\sum_{k=0}^{n}\left(-1\right)^{n+k}S_{1}\left(n,k\right)x^{k} \end{align*} となるので、係数を比較すると、\(S_{1}'\left(n,k\right)=\left(-1\right)^{n+k}S_{1}\left(n,k\right)\)となる。
\[ Q\left(x,n\right)=\sum_{k=0}^{n}S_{1}'\left(n,k\right)x^{k} \] で定義する場合もある。
この定義と、
\[ P\left(x,n\right)=\sum_{k=0}^{n}S_{1}\left(n,k\right)x^{k} \] で定義した場合と比べる。
これらの第1種スターリング数同士の関係は、
\begin{align*} \sum_{k=0}^{n}S_{1}'\left(n,k\right)x^{k} & =Q\left(x,n\right)\\ & =\left(-1\right)^{n}P\left(-x,n\right)\\ & =\left(-1\right)^{n}\sum_{k=0}^{n}S_{1}\left(n,k\right)\left(-x\right)^{k}\\ & =\sum_{k=0}^{n}\left(-1\right)^{n+k}S_{1}\left(n,k\right)x^{k} \end{align*} となるので、係数を比較すると、\(S_{1}'\left(n,k\right)=\left(-1\right)^{n+k}S_{1}\left(n,k\right)\)となる。
-
\(S_{1}\left(3,0\right),S_{1}\left(3,1\right),S_{1}\left(3,2\right),S_{1}\left(3,3\right)\)を求めてみる。\[ P\left(x,3\right)=\sum_{k=0}^{3}S_{1}\left(3,k\right)x^{k} \] なので左辺は、
\begin{align*} P\left(x,3\right) & =x\left(x-1\right)\left(x-2\right)\\ & =x^{3}-3x^{2}+2x \end{align*} 右辺は、
\[ \sum_{k=0}^{3}S_{1}\left(3,k\right)x^{k}=S_{1}\left(3,0\right)+S_{1}\left(3,1\right)x+S_{1}\left(3,2\right)x^{2}+S_{1}\left(3,3\right)x^{3} \] となるので、
\[ x^{3}-3x^{2}+2x=S_{1}\left(3,3\right)x^{3}+S_{1}\left(3,2\right)x^{2}+S_{1}\left(3,1\right)x+S_{1}\left(3,0\right) \] より、\(x\)の係数を比較すると、
\[ \begin{cases} S_{1}\left(3,0\right)=0\\ S_{1}\left(3,1\right)=2\\ S_{1}\left(3,2\right)=-3\\ S_{1}\left(3,3\right)=1 \end{cases} \] となる。
-
\(S_{2}\left(3,0\right),S_{2}\left(3,1\right),S_{2}\left(3,2\right),S_{2}\left(3,3\right)\)を求めてみる。\begin{align*} x^{3} & =\sum_{k=0}^{3}S_{2}\left(3,k\right)P\left(x,k\right)\\ & =S_{2}\left(3,0\right)P\left(x,0\right)+S_{2}\left(3,1\right)P\left(x,1\right)+S_{2}\left(3,2\right)P\left(x,2\right)+S_{2}\left(3,3\right)P\left(x,3\right)\\ & =S_{2}\left(3,0\right)+S_{2}\left(3,1\right)x+S_{2}\left(3,2\right)x\left(x-1\right)+S_{2}\left(3,3\right)x\left(x-1\right)\left(x-2\right)\\ & =S_{2}\left(3,0\right)+S_{2}\left(3,1\right)x+S_{2}\left(3,2\right)\left(x^{2}-x\right)+S_{2}\left(3,3\right)\left(x^{3}-3x^{2}+2x\right)\\ & =S_{2}\left(3,3\right)x^{3}+\left\{ S_{2}\left(3,2\right)-3S_{2}\left(3,3\right)\right\} x^{2}+\left\{ S_{2}\left(3,1\right)-S_{2}\left(3,2\right)+2S_{2}\left(3,3\right)\right\} x+S_{2}\left(3,0\right) \end{align*} より、\(x\)の係数を比較すると、
\[ \begin{cases} S_{2}\left(3,0\right)=0\\ S_{2}\left(3,1\right)-S_{2}\left(3,2\right)+2S_{2}\left(3,3\right)=0\\ S_{2}\left(3,2\right)-3S_{2}\left(3,3\right)=0\\ S_{2}\left(3,3\right)=1 \end{cases} \] となる。
これを解くと、
\[ \begin{cases} S_{2}\left(3,0\right)=0\\ S_{2}\left(3,1\right)=1\\ S_{2}\left(3,2\right)=3\\ S_{2}\left(3,3\right)=1 \end{cases} \] となる。
ページ情報
| タイトル | 第1種スターリング数と第2種スターリング数の定義 |
| URL | https://www.nomuramath.com/cnrigdyh/ |
| SNSボタン |
スターリング数とベルヌーイ数の関係
\[
\frac{\left(-1\right)^{m}}{m!}\sum_{k=0}^{m}\left(-1\right)^{k}S_{1}\left(m+1,k+1\right)B_{k}=\frac{1}{m+1}
\]
第1種スターリング数の符号
\[
\left|S_{1}\left(n,k\right)\right|=\left(-1\right)^{n+k}S_{1}\left(n,k\right)
\]
第2種スターリング数の一般解
\[
S_{2}\left(n,k\right)=\frac{1}{k!}\sum_{j=0}^{k}\left(-1\right)^{k-j}C\left(k,j\right)j^{n}
\]
(*)スターリング数と2項係数
\[
C\left(k,m\right)S_{1}\left(n,k\right)=\sum_{j=k-m}^{n-m}C\left(n,j\right)S_{1}\left(n-j,m\right)S_{1}\left(j,k-m\right),m\leq k
\]