中央2項係数の総和

\begin{align*} \sum_{k=0}^{\infty}C^{-1}\left(2k,k\right) & =\sum_{k=0}^{\infty}\frac{(k!)^{2}}{(2k)!}\\ & =\sum_{k=0}^{\infty}\frac{\Gamma(k+1)\Gamma(k+1)}{\Gamma(2k+1)}\\ & =1+\sum_{k=1}^{\infty}\frac{\Gamma(k+1)\Gamma(k+1)}{\Gamma(2k+1)}\\ & =1+\sum_{k=1}^{\infty}\frac{k\Gamma(k)\Gamma(k+1)}{\Gamma(2k+1)}\\ & =1+\sum_{k=1}^{\infty}kB(k,k+1)\\ & =1+\sum_{k=1}^{\infty}k\int_{0}^{1}t^{k-1}(1-t)^{k}dt\\ & =1+\sum_{k=1}^{\infty}k\int_{0}^{1}\frac{1}{t}(t-t^{2})^{k}dt\\ & =1+\sum_{k=1}^{\infty}\int_{0}^{1}\frac{1}{t}\left(t-t^{2}\right)\frac{d}{d\left(t-t^{2}\right)}(t-t^{2})^{k}dt\\ & =1+\int_{0}^{1}\frac{1}{t}\left(t-t^{2}\right)\frac{d}{d\left(t-t^{2}\right)}\frac{\left(t-t^{2}\right)}{1-(t-t^{2})}dt\\ & =1+\int_{0}^{1}\frac{1}{t}\left(t-t^{2}\right)\frac{1}{\left((t-t^{2})-1\right)^{2}}dt\\ & =1+\int_{0}^{1}\frac{1-t}{\left(t^{2}-t+1\right)^{2}}dt\\ & =1+\int_{0}^{1}\frac{1-t}{\left(t^{2}-t+1\right)^{2}}dt\\ & =1+\int_{0}^{1}\frac{\frac{1}{2}-\left(t-\frac{1}{2}\right)}{\left(t-\frac{1}{2}\right)^{2}+\frac{3}{4}}dt\\ & =1-\frac{4}{9}\int_{-\frac{1}{\sqrt{3}}}^{\frac{1}{\sqrt{3}}}\frac{3x-\sqrt{3}}{\left(x^{2}+1\right)^{2}}dx\qquad,\qquad\frac{\sqrt{3}}{2}x=\left(t-\frac{1}{2}\right)\\ & =1-\frac{4}{9}\int_{-\frac{\pi}{6}}^{\frac{\pi}{6}}\frac{3\tan u-\sqrt{3}}{\left(\tan^{2}u+1\right)^{2}}\frac{1}{\cos^{2}u}du\qquad,\qquad x=\tan u\\ & =1-\frac{4}{9}\int_{-\frac{\pi}{6}}^{\frac{\pi}{6}}\left(3\sin u\cos u-\sqrt{3}\cos^{2}u\right)du\\ & =1-\frac{4}{9}\int_{-\frac{\pi}{6}}^{\frac{\pi}{6}}\left(\frac{3}{2}\sin\left(2u\right)-\frac{\sqrt{3}}{2}\left(\cos\left(2u\right)+1\right)\right)du\\ & =1-\frac{4}{9}\left[-\frac{3}{4}\cos\left(2u\right)-\frac{\sqrt{3}}{4}\sin\left(2u\right)-\frac{\sqrt{3}}{2}u\right]_{-\frac{\pi}{6}}^{\frac{\pi}{6}}\\ & =1-\frac{4}{9}\left[-\frac{\sqrt{3}}{2}\sin\left(2u\right)-\sqrt{3}u\right]_{0}^{\frac{\pi}{6}}\\ & =\frac{4}{3}+\frac{2\sqrt{3}\pi}{27} \end{align*}