２項係数

\[\sum_{k=1}^{n-1}\frac{C\left(k-n,k\right)}{k}=-H_{n-1}\]

\[C\left(\alpha,\beta\right)C\left(\beta,\gamma\right)=C\left(\alpha,\gamma\right)C\left(\alpha-\gamma,\beta-\gamma\right)\]

\[\sum_{k_{1}+\cdots+k_{p}=m}\prod_{j=1}^{p}C\left(n_{j},k_{j}\right)=C\left(\sum_{j=1}^{p}n_{j},m\right)\]

\[\sum_{j=0}^{k-a}\left(-1\right)^{j}C\left(k,j+a\right)C\left(j+b,c\right)=\begin{cases}\left(-1\right)^{k-a}C\left(b-a,c-k\right) & a-b+c\leq k\\0 & k<a-b+c\end{cases}\]

\[\sum_{k=0}^{\infty}b_{k}x^{k}=\frac{1}{1-x}\sum_{k=0}^{\infty}a_{k}\left(\frac{x}{1-x}\right)^{k}\]

\[\sum_{k=0}^{\infty}C\left(n,2k\right)a^{2k}b^{n-2k}=\frac{1}{2}\left\{ \left(a+b\right)^{n}+\left(-a+b\right)^{n}\right\} \]

\[\sum_{k=-\infty}^{\infty}C\left(mn,mk+l\right)=\frac{1}{m}\sum_{j=0}^{m-1}\left(1+\omega_{m}^{j}\right)^{mn}\left(\omega_{m}^{j}\right)^{-l}\]

\[\forall m,n\in\mathbb{Z},\left(0\leq m<n\right)\lor\left(n<0\leq m\right)\lor\left(m<n<0\right)\Leftrightarrow C\left(m,n\right)=0\]

\[\sum_{k=0}^{n}\frac{\left(-1\right)^{k}C\left(n,k\right)}{m+k}=\frac{1}{mC\left(m+n,m\right)}\]

\[C\left(2n,n\right)=4^{n}\left(-1\right)^{n}C\left(-\frac{1}{2},n\right)\]

\[C\left(-m,-n\right)=\left(-1\right)^{m-n}C\left(n-1,m-1\right)\]

\[C\left(x+n,y+n\right)=\sum_{k=0}^{n}C\left(n,k\right)C\left(x,y+k\right)\]

\[C\left(x+n,y+n\right)=C\left(x,y+n\right)+\sum_{k=0}^{n-1}C\left(x+k,y+n-1\right)\]

\[\sum_{k=0}^{\infty}\frac{1}{k+1}C\left(2k,k\right)z^{k}=\frac{1}{2z}\left\{ 1-\left(1-4z\right)^{\frac{1}{2}}\right\} \]

\[\sum_{k=0}^{\infty}C\left(2k,k\right)z^{k}=\left(1-4z\right)^{-\frac{1}{2}}\]

\[\sum_{k=0}^{n-1}C\left(2n-1,k\right)=2^{2n-2}\]

\[\lim_{n\rightarrow\infty}\sqrt[n]{\sqrt[n+1]{\prod_{k=0}^{n}C\left(n,k\right)}}=\sqrt{e}\]

\[a_{n}=\sum_{k=0}^{n}(-1)^{n-k}C(n,k)b_{k}\]

\[\sum_{k=0}^{\infty}C(x+k,k)t^{k}=(1-t)^{-(x+1)}\]

\[\frac{d}{dx}C(x,y) =C(x,y)\left(\psi(1+x)-\psi(1+x-y)\right)\]

\[C^{-1}(k+j+1,j+1)=\frac{j+1}{j}\left(C^{-1}(k+j,j)-C^{-1}(k+j+1,j)\right)\]

\[\sum_{k=-a}^{a}(-1)^{k}C(a+b,a+k)C(b+c,b+k)C(c+a,c+k)=\frac{(a+b+c)!}{a!b!c!}\]

\[ \sum_{j=0}^{m}C^{2}(m,j)=C(2m,m)\]

\[ \sum_{j=0}^{k}C(x,j)C(y,k-j)=C(x+y,k)\]

\[ C(x,t)C(t,y)=C(x,y)C(x-y,x-t)\]

\[\sum_{k=0}^{n}P(k,m)C(n,k)=P(n,m)2^{n-m}\]

\[ C(x+1,y+1)=C(x,y+1)+C(x,y)\]

\[ C(x+1,y)=\frac{x+1}{x+1-y}C(x,y)\]