\begin{align*} \sum_{k=0}^{\infty}C_{k}x^{k} & =1+\sum_{k=1}^{\infty}C_{k}x^{k}\\ & =1+\sum_{k=0}^{\infty}C_{k+1}x^{k+1}\\ & =1+\sum_{k=0}^{\infty}\sum_{j=0}^{k}C_{j}C_{k-j}x^{k+1}\\ & =1+\sum_{j=0}^{\infty}\sum_{k=j}^{\infty}C_{j}C_{k-j}x^{k+1}\\ & =1+\sum_{j=0}^{\infty}\sum_{k=0}^{\infty}C_{j}C_{k}x^{k+j+1}\\ & =1+x\sum_{j=0}^{\infty}C_{j}x^{j}\sum_{k=0}^{\infty}C_{k}x^{k}\\ & =1+x\LHS^{2}\\ & =\frac{1\pm\sqrt{1-4x}}{2x}\\ & =\frac{1-\sqrt{1-4x}}{2x}\cmt{x\rightarrow0\text{で上側の符号は不適}} \end{align*}