\[ a_{n}=\sum_{k=0}^{\left\lfloor \frac{n}{2}\right\rfloor }C\left(n,2k\right)b_{n-2k} \] であるとき、
\begin{align*} \sum_{k=0}^{\left\lfloor \frac{n}{2}\right\rfloor }C\left(n,2k\right)E_{2k}a_{n-2k} & =\sum_{k=0}^{\left\lfloor \frac{n}{2}\right\rfloor }C\left(n,2k\right)E_{2k}\sum_{j=0}^{\left\lfloor \frac{n}{2}\right\rfloor -k}C\left(n-2k,2j\right)b_{n-2k-2j}\\ & =\sum_{k=0}^{\left\lfloor \frac{n}{2}\right\rfloor }C\left(n,2k\right)E_{2k}\sum_{j=k}^{\left\lfloor \frac{n}{2}\right\rfloor }C\left(n-2k,2j-2k\right)b_{n-2j}\\ & =\sum_{j=0}^{\left\lfloor \frac{n}{2}\right\rfloor }\sum_{k=0}^{j}C\left(n,2k\right)E_{2k}C\left(n-2k,2j-2k\right)b_{n-2j}\\ & =\sum_{j=0}^{\left\lfloor \frac{n}{2}\right\rfloor }\sum_{k=0}^{\left\lfloor \frac{n}{2}\right\rfloor -j}C\left(n,2k\right)E_{2k}C\left(n-2k,2\left\lfloor \frac{n}{2}\right\rfloor -2j-2k\right)b_{n-2\left(\left\lfloor \frac{n}{2}\right\rfloor -j\right)}\cmt{j\rightarrow\left\lfloor \frac{n}{2}\right\rfloor -j}\\ & =\sum_{j=0}^{\left\lfloor \frac{n}{2}\right\rfloor }\sum_{k=0}^{\left\lfloor \frac{n}{2}\right\rfloor -j}C\left(n,2k\right)E_{2k}C\left(n-2k,n-2\left\lfloor \frac{n}{2}\right\rfloor +2j\right)b_{n-2\left(\left\lfloor \frac{n}{2}\right\rfloor -j\right)}\\ & =\sum_{j=0}^{\left\lfloor \frac{n}{2}\right\rfloor }\sum_{k=0}^{\left\lfloor \frac{n}{2}\right\rfloor -j}C\left(n,2\left\lfloor \frac{n}{2}\right\rfloor -2j\right)E_{2k}C\left(2\left\lfloor \frac{n}{2}\right\rfloor -2j,2k\right)b_{n-2\left(\left\lfloor \frac{n}{2}\right\rfloor -j\right)}\cmt{\because C\left(a,b\right)C\left(b,c\right)=C\left(a,c\right)C\left(a-c,a-b\right)}\\ & =\sum_{j=0}^{\left\lfloor \frac{n}{2}\right\rfloor }C\left(n,2\left\lfloor \frac{n}{2}\right\rfloor -2j\right)b_{n-2\left(\left\lfloor \frac{n}{2}\right\rfloor -j\right)}\sum_{k=0}^{\left\lfloor \frac{n}{2}\right\rfloor -j}C\left(2\left(\left\lfloor \frac{n}{2}\right\rfloor -j\right),2k\right)E_{2k}\\ & =\sum_{j=0}^{\left\lfloor \frac{n}{2}\right\rfloor }C\left(n,2\left\lfloor \frac{n}{2}\right\rfloor -2j\right)b_{n-2\left(\left\lfloor \frac{n}{2}\right\rfloor -j\right)}\delta_{0,\left\lfloor \frac{n}{2}\right\rfloor -j}\cmt{\delta_{0,n}=\sum_{k=0}^{n}C\left(2j,2k\right)E_{2k}}\\ & =C\left(n,0\right)b_{n}\\ & =b_{n} \end{align*} となる。
従って題意は成り立つ。

-

逆向きは次のようにする。
\begin{align*} \sum_{k=0}^{\left\lfloor \frac{n}{2}\right\rfloor }C\left(n,2k\right)b_{n-2k} & =\sum_{k=0}^{\left\lfloor \frac{n}{2}\right\rfloor }C\left(n,2k\right)\sum_{j=0}^{\left\lfloor \frac{n-2k}{2}\right\rfloor }C\left(n-2k,2j\right)E_{2j}a_{n-2k-2j}\\ & =\sum_{k=0}^{\left\lfloor \frac{n}{2}\right\rfloor }C\left(n,2k\right)\sum_{j=k}^{\left\lfloor \frac{n}{2}\right\rfloor }C\left(n-2k,2j-2k\right)E_{2j-2k}a_{n-2j}\\ & =\sum_{j=0}^{\left\lfloor \frac{n}{2}\right\rfloor }\sum_{k=0}^{j}C\left(n,2k\right)C\left(n-2k,2j-2k\right)E_{2j-2k}a_{n-2j}\\ & =\sum_{j=0}^{\left\lfloor \frac{n}{2}\right\rfloor }\sum_{k=0}^{\left\lfloor \frac{n}{2}\right\rfloor -j}C\left(n,2k\right)C\left(n-2k,2\left\lfloor \frac{n}{2}\right\rfloor -2j-2k\right)E_{2\left\lfloor \frac{n}{2}\right\rfloor -2j-2k}a_{n-2\left\lfloor \frac{n}{2}\right\rfloor -2j}\cmt{j\rightarrow\left\lfloor \frac{n}{2}\right\rfloor -j}\\ & =\sum_{j=0}^{\left\lfloor \frac{n}{2}\right\rfloor }\sum_{k=0}^{\left\lfloor \frac{n}{2}\right\rfloor -j}C\left(n,2k\right)C\left(n-2k,n-2\left\lfloor \frac{n}{2}\right\rfloor +2j\right)E_{2\left\lfloor \frac{n}{2}\right\rfloor -2j-2k}a_{n-2\left\lfloor \frac{n}{2}\right\rfloor -2j}\\ & =\sum_{j=0}^{\left\lfloor \frac{n}{2}\right\rfloor }\sum_{k=0}^{\left\lfloor \frac{n}{2}\right\rfloor -j}C\left(n,2\left\lfloor \frac{n}{2}\right\rfloor -2j\right)C\left(2\left\lfloor \frac{n}{2}\right\rfloor -2j,2k\right)E_{2\left\lfloor \frac{n}{2}\right\rfloor -2j-2k}a_{n-2\left\lfloor \frac{n}{2}\right\rfloor -2j}\cmt{\because C\left(a,b\right)C\left(b,c\right)=C\left(a,c\right)C\left(a-c,a-b\right)}\\ & =\sum_{j=0}^{\left\lfloor \frac{n}{2}\right\rfloor }\sum_{k=0}^{\left\lfloor \frac{n}{2}\right\rfloor -j}C\left(n,2\left\lfloor \frac{n}{2}\right\rfloor -2j\right)C\left(2\left(\left\lfloor \frac{n}{2}\right\rfloor -j\right),2k\right)E_{2\left(\left\lfloor \frac{n}{2}\right\rfloor -j-k\right)}a_{n-2\left\lfloor \frac{n}{2}\right\rfloor -2j}\\ & =\sum_{j=0}^{\left\lfloor \frac{n}{2}\right\rfloor }C\left(n,2\left\lfloor \frac{n}{2}\right\rfloor -2j\right)a_{n-2\left\lfloor \frac{n}{2}\right\rfloor -2j}\sum_{k=0}^{\left\lfloor \frac{n}{2}\right\rfloor -j}C\left(2\left(\left\lfloor \frac{n}{2}\right\rfloor -j\right),2\left(\left\lfloor \frac{n}{2}\right\rfloor -j-k\right)\right)E_{2k}\\ & =\sum_{j=0}^{\left\lfloor \frac{n}{2}\right\rfloor }C\left(n,2\left\lfloor \frac{n}{2}\right\rfloor -2j\right)a_{n-2\left\lfloor \frac{n}{2}\right\rfloor -2j}\sum_{k=0}^{\left\lfloor \frac{n}{2}\right\rfloor -j}C\left(2\left(\left\lfloor \frac{n}{2}\right\rfloor -j\right),2k\right)E_{2k}\\ & =\sum_{j=0}^{\left\lfloor \frac{n}{2}\right\rfloor }C\left(n,2\left\lfloor \frac{n}{2}\right\rfloor -2j\right)a_{n-2\left\lfloor \frac{n}{2}\right\rfloor -2j}\delta_{j,\left\lfloor \frac{n}{2}\right\rfloor }\cmt{\delta_{0,n}=\sum_{k=0}^{n}C\left(2j,2k\right)E_{2k}}\\ & =C\left(n,0\right)a_{n}\\ & =a_{n} \end{align*}