[组合] 收集组合恒等式

刚才在超级群看到有人提到“组合恒等式表”，我手头上没有这类资料，不如就在这里收集一些吧

群里看到的题是：

（1）$C_{2n}^0+C_{2n-1}^1+C_{2n-2}^2+\cdots+C_n^n=?$

右边待填，暂无答案。

再列几个以前看到过的：

（2）$C_n^1+2C_n^2+\cdots+nC_n^n=n2^{n-1}$

（3）$(C_n^1)^2+(C_n^2)^2+\cdots+(C_n^n)^2=C_{2n}^n$

（3’）$C_m^0C_n^k+C_m^1C_n^{k-1}+\cdots+C_m^kC_n^0=C_{m+n}^k$

（4）$\displaystyle 1-\frac12C_n^1+\frac13C_n^2+\cdots+(-1)^n\frac1{n+1}C_n^n=\frac1{n+1}$

（5）$\displaystyle C_n^1-\frac{C_n^2}2+\cdots+\frac{(-1)^{n-1}C_n^n}n=1+\frac12+\cdots+\frac1n$

到你们了

K版，等哪天有空我把史济怀那本组合恒等式的附录编辑成latex吧。。

2# Tesla35

直接在这里编辑也行

今天先写前十个

$\displaystyle C_n^k=C_n^{n-k}$
$\displaystyle C_n^k=C_{n-1}^k+C_{n-1}^{k-1}$
$\displaystyle C_n^k=\frac{n}{k}C_{n-1}^{k-1}$
$\displaystyle C_n^kC_k^m=C_n^mC_{n-m}^{k-m}=C_n^{k-m}C_{n-k+m}^m(m\leqslant k\leqslant n)$
$\displaystyle \sum_{k=0}^{n}C_n^k=2^n$
$\displaystyle \sum_{k=0}^{n}(-1)^kC_n^k=0$
$\displaystyle \sum_{k=0}^{m}(-1)^kC_n^k=(-1)^mC_{n-1}^m(m<n)$
$\displaystyle \sum_{k=0}^{n}C_{2n}^k=2^{n-1}+\frac{1}{2}C_{2n}^n$
$\displaystyle \sum_{k=0}^{n}\frac{1}{k+1}C_n^k=\frac{1}{n+1}(2^{n+1}-1)$
$\displaystyle \sum_{k=0}^{n}(-1)^k\frac{1}{k+1}C_n^k=\frac{1}{n+1}$

擦用不了numerate么。。

4# Tesla35

oh，的确用不了，既然如此，还是在真 latex 中写吧

前41个恒等式
\item $\displaystyle C_n^k=C_n^{n-k}$
\item $\displaystyle C_n^k=C_{n-1}^k+C_{n-1}^{k-1}$
\item $\displaystyle C_n^k=\frac{n}{k}C_{n-1}^{k-1}$
\item $\displaystyle C_n^kC_k^m=C_n^mC_{n-m}^{k-m}=C_n^{k-m}C_{n-k+m}^m(m\leqslant k\leqslant n)$
\item $\displaystyle \sum_{k=0}^{n}C_n^k=2^n$
\item $\displaystyle \sum_{k=0}^{n}(-1)^kC_n^k=0$
\item $\displaystyle \sum_{k=0}^{m}(-1)^kC_n^k=(-1)^mC_{n-1}^m(m<n)$
\item $\displaystyle \sum_{k=0}^{n}C_{2n}^k=2^{n-1}+\frac{1}{2}C_{2n}^n$
\item $\displaystyle \sum_{k=0}^{n}\frac{1}{k+1}C_n^k=\frac{1}{n+1}(2^{n+1}-1)$
\item $\displaystyle \sum_{k=0}^{n}(-1)^k\frac{1}{k+1}C_n^k=\frac{1}{n+1}$ %10
\item $\displaystyle \sum_{k=1}^{n}k^2C_n^k=n(n+1)2^{n-2}$
\item $\displaystyle \sum_{k=0}^{n}(-1)^kC_n^k\frac{m}{m+k}=(C_{m+n}^n)^{-1}$
\item $\displaystyle \sum_{k=1}^{n}(-1)^{k+1}\frac{1}{k}C_n^k=1+\frac{1}{2}+\cdots+\frac{1}{n}$
\item $\displaystyle \sum_{k=0}^{n}(-1)^k2^{2n-2k}C_{2n-k+1}^k=n+1$
\item $\displaystyle \sum_{k=0}^{n}(-1)^k2^{2n-2k}C_{2n-k}^k=2n+1$
\item $\displaystyle \sum_{k=0}^{n}2^kC_n^k=3^n$
\item $\displaystyle \sum_{k=0}^{n}kC_n^k=n2^{n-1}$
\item $\displaystyle \sum_{k=0}^{n}(k+1)C_n^k=(n+2)2^{n-1}$
\item $\displaystyle \sum_{k=0}^{n}(-1)^kkC_n^k=0(n>1)$
\item $\displaystyle \sum_{k=0}^{n}\frac{1}{k+1}C_n^kx^k=\frac{(1+x)^{n+1}-1}{(n+1)x}$%20
\item $\displaystyle \sum_{k=0}^{n}\frac{(-1)^k}{(k+1)^2}C_n^k=\frac{1}{n+1}\left(1+\frac{1}{2}+\cdots+\frac{1}{n+1}\right)$
\item $\displaystyle \sum_{k=0}^{n-1}C_{2n-1}^k=2^{2n-2}$
\item $\displaystyle \sum_{k=0}^{n}kC_{2n}^k=n2^{2n-1}$
\item $\displaystyle \sum_{k=1}^{n}(-1)^{k+1}\frac{1}{k}C_n^k[1-(1-x)^k]=\sum_{k=1}^{n}\frac{x^k}{k}$
\item $\displaystyle \sum_{k=0}^{n}\frac{(-1)^k}{2k+1}C_n^k=\frac{2^{2n}}{2n+1}(C_{2n}^n)^{-1}$
\item $\displaystyle \sum_{k=0}^{n}\frac{(-1)^k}{1-2k}C_n^k=2^{2n}(C_{2n}^n)^{-1}$
\item $\displaystyle \sum_{k=0}^{n}kC_{2n+1}^k=(2n+1)2^{2n-1}-\frac{2n+1}{2}C_{2n}^n$
\item $\displaystyle \sum_{k=0}^{n}kC_{2n}^{n-k}=nC_{2n-1}^{n}$
\item $\displaystyle \sum_{k=0}^{n}kC_{2n+1}^{n-k}=(2n+1)C_{2n-1}^{n}-2^{2n-1}$
\item $\displaystyle \sum_{k=0}^{n}C_{2n+1}^{2k+1}x^k=2x^{-\frac{1}{2}}[(1+\sqrt{x})^{2n+1}-(1-\sqrt{x})^{2n+1}]$
\item $\displaystyle \sum_{k=0}^{[\frac{n}{2}]}C_{n-k}^k=
\frac{1}{\sqrt{5}}\left\{\left(\frac{1+\sqrt{5}}{2}\right)^{n+1}-\left(\frac{1-\sqrt{5}}{2}\right)^{n+1}\right\}$
\item $\displaystyle \sum_{k=0}^{[\frac{n}{2}]}C_{n-k}^kr^k=
\frac{1}{2^{n+1}\sqrt{1+4r}}\left\{(1+\sqrt{1+4r})^{n+1}-(1-\sqrt{1+4r})^{n+1}\right\}$
\item $\displaystyle \sum_{k=0}^{[\frac{n}{2}]}C_{n-k}^k2^k=\frac{1}{3}(2^{n+1}+(-1)^n)$
\item $\displaystyle \sum_{k=0}^{[\frac{n}{2}]}C_{n-k}^k6^k=\frac{1}{5}(3^{n+1}+(-1)^n2^{n+1})$
\item $\displaystyle \sum_{k=0}^{[\frac{n}{2}]}(-1)^kC_{n-k}^k=\frac{2}{\sqrt{3}}\sin\frac{n+1}{3}\pi$
\item $\displaystyle \sum_{k=0}^{n-1}C_{n+k}^{2k+1}=
\frac{1}{\sqrt{5}}\left\{\left(\frac{3+\sqrt{5}}{2}\right)^{n}-\left(\frac{3-\sqrt{5}}{2}\right)^{n}\right\}$
\item $\displaystyle \sum_{k=0}^{n}C_{n+k}^{2k}=
\frac{1}{\sqrt{5}}\left\{\left(\frac{1+\sqrt{5}}{2}\right)^{2n+1}-\left(\frac{1-\sqrt{5}}{2}\right)^{2n+1}\right\}$
\item $\displaystyle \sum_{k=0}^{m}C_{k+n-1}^{n-1}=C_{m+n}^{n}$
\item $\displaystyle \sum_{k=0}^{n}2^{n-k}C_{n+k}^{2k}=\frac{1}{3}(2^{2n+1}+1)$
\item $\displaystyle \sum_{k=0}^{n}(-1)^kC_n^k\left(1+\frac{1}{2}+\cdots+\frac{1}{k}\right)=-\frac{1}{n}$
\item $\displaystyle \sum_{k=0}^{n}(-1)^kC_n^k\left(x+\frac{x^2}{2}+\cdots+\frac{x^k}{k}\right)=-\frac{1}{n}[1-(1-x)^n]$

这么叉多……果然够表……

第41~60个
\item $\displaystyle \sum_{k=0}^{n}(-1)^kC_n^k\left(x+\frac{x^2}{2}+\cdots+\frac{x^k}{k}\right)=-\frac{1}{n}[1-(1-x)^n]$
\item $\displaystyle \sum_{k=0}^{n}C_{n+p}^{k+p}x^k=\sum_{k=0}^{n}C_{p+k-1}^n(1+x)^{n-k}$
\item $\displaystyle \sum_{k=0}^{n}(-1)^kC_{n+p}^{k+p}=C_{n+p-1}^n$
\item $\displaystyle \sum_{k=0}^{[\frac{n}{2}]}(-1)^k\frac{n}{n-k}C_{n-k}^kx^{n-2k}=
\frac{1}{2^n}\left\{(x+\sqrt{x^2-4})^n+(x-\sqrt{x^2-4})^n\right\}$
\item $\displaystyle \sum_{k=0}^{[\frac{n}{2}]}(-1)^k\frac{n}{n-k}C_{n-k}^k2^{n-2k}=2$
\item $\displaystyle \sum_{k=0}^{[\frac{n}{2}]}(-1)^k\frac{n}{n-k}C_{n-k}^k=2\cos\frac{n\pi}{3},n=1,2,\cdots.$
\item $\displaystyle \sum_{k=0}^{[\frac{n}{2}]}C_n^k\cos\frac{(n-2k)}{3}\pi
=\left\{ {\begin{array}{*{20}{c}}
{\displaystyle\frac{1}{2},n\text{为奇数}}\\
{\displaystyle\frac{1}{2}(1+C_n^{\frac{n}{2}}),n\text{为偶数}}
\end{array}} \right.$
\item $\displaystyle \sum_{k=0}^{[\frac{n}{2}]}\frac{(n+1-2k)^2}{n+1-k}C_n^k=2^n$
\item $\displaystyle \sum_{k=0}^{n}\frac{(-1)^k}{k+1}C_n^k[(1+x)^{k+1}-1]=(-1)^n\frac{x^{n+1}}{n+1}$
\item $\displaystyle \sum_{k=0}^{n}\frac{(-1)^k}{k+1}C_n^k\left(1+\frac{1}{2}+\cdots+\frac{1}{k+1}\right)=\frac{1}{(n+1)^2}$%50
\item $\displaystyle \sum_{k=1}^{n}(-1)^{k+1}C_n^k\sum_{l=1}^k\frac{1}{l}\left(1+\frac{1}{2}+\cdots+\frac{1}{l}\right)=\frac{1}{n^2}$
\item $\displaystyle \sum_{k=0}^{n}\frac{(-1)^{k}}{k+1}C_n^k\sum_{l=0}^k\frac{1}{l+1}\left(1+\frac{1}{2}+\cdots+\frac{1}{l+1}\right)
    =\frac{1}{(n+1)^3}$
\item $\displaystyle \sum_{k=0}^{[\frac{n}{2}]}(-1)^kC_{n-k}^k2^{n-2k}=n+1$
\item $\displaystyle \sum_{k=0}^{[\frac{n}{2}]}(-1)^kC_{n-k}^k\frac{(2\cos x)^{n-2k}}{n-k}=\frac{2}{n}\cos nx$
\item $\displaystyle \sum_{k=0}^{[\frac{n}{2}]}C_n^k\cos(n-2k)x
=\left\{ {\begin{array}{*{20}{c}}
{\displaystyle 2^{n-1}\cos^nx,n\text{为奇数}}\\
{\displaystyle 2^{n-1}\cos^nx+\frac{1}{2}C_n^{\frac{n}{2}},n\text{为偶数}}
\end{array}} \right.$
\item $\displaystyle \sum_{k=1}^{n}\frac{(-1)^{k+1}}{k}C_n^k\left(1+\frac{1}{2}+\cdots+\frac{1}{k}\right)=\sum_{k=1}^n\frac{1}{k^2}$
\item $\displaystyle \sum_{k=1}^{n}(-1)^{k}C_n^k\left(1+\frac{1}{2^2}+\cdots+\frac{1}{k^2}\right)=-\frac{1}{n}\sum_{k=1}^n\frac{1}{k}$
\item $\displaystyle \sum_{k=0}^{n}(-1)^kC_n^k(x+n-k)^n=n!$
\item $\displaystyle \sum_{k=0}^{n}(-1)^kP(k)C_n^k=0$这里$P(k)$是$k$的$p(<n)$次多项式
\item $\displaystyle \sum_{k=0}^{n}(-1)^kC_n^k(x-k)^{n+1}=\left(x-\frac{n}{2}\right)(n+1)!$%60
还有100个

niubility

1~160:（全）
\item $\displaystyle C_n^k=C_n^{n-k}$
\item $\displaystyle C_n^k=C_{n-1}^k+C_{n-1}^{k-1}$
\item $\displaystyle C_n^k=\frac{n}{k}C_{n-1}^{k-1}$
\item $\displaystyle C_n^kC_k^m=C_n^mC_{n-m}^{k-m}=C_n^{k-m}C_{n-k+m}^m(m\leqslant k\leqslant n)$
\item $\displaystyle \sum_{k=0}^{n}C_n^k=2^n$
\item $\displaystyle \sum_{k=0}^{n}(-1)^kC_n^k=0$
\item $\displaystyle \sum_{k=0}^{m}(-1)^kC_n^k=(-1)^mC_{n-1}^m(m<n)$
\item $\displaystyle \sum_{k=0}^{n}C_{2n}^k=2^{n-1}+\frac{1}{2}C_{2n}^n$
\item $\displaystyle \sum_{k=0}^{n}\frac{1}{k+1}C_n^k=\frac{1}{n+1}(2^{n+1}-1)$
\item $\displaystyle \sum_{k=0}^{n}(-1)^k\frac{1}{k+1}C_n^k=\frac{1}{n+1}$ %10
\item $\displaystyle \sum_{k=1}^{n}k^2C_n^k=n(n+1)2^{n-2}$
\item $\displaystyle \sum_{k=0}^{n}(-1)^kC_n^k\frac{m}{m+k}=(C_{m+n}^n)^{-1}$
\item $\displaystyle \sum_{k=1}^{n}(-1)^{k+1}\frac{1}{k}C_n^k=1+\frac{1}{2}+\cdots+\frac{1}{n}$
\item $\displaystyle \sum_{k=0}^{n}(-1)^k2^{2n-2k}C_{2n-k+1}^k=n+1$
\item $\displaystyle \sum_{k=0}^{n}(-1)^k2^{2n-2k}C_{2n-k}^k=2n+1$
\item $\displaystyle \sum_{k=0}^{n}2^kC_n^k=3^n$
\item $\displaystyle \sum_{k=0}^{n}kC_n^k=n2^{n-1}$
\item $\displaystyle \sum_{k=0}^{n}(k+1)C_n^k=(n+2)2^{n-1}$
\item $\displaystyle \sum_{k=0}^{n}(-1)^kkC_n^k=0(n>1)$
\item $\displaystyle \sum_{k=0}^{n}\frac{1}{k+1}C_n^kx^k=\frac{(1+x)^{n+1}-1}{(n+1)x}$%20
\item $\displaystyle \sum_{k=0}^{n}\frac{(-1)^k}{(k+1)^2}C_n^k=\frac{1}{n+1}\left(1+\frac{1}{2}+\cdots+\frac{1}{n+1}\right)$
\item $\displaystyle \sum_{k=0}^{n-1}C_{2n-1}^k=2^{2n-2}$
\item $\displaystyle \sum_{k=0}^{n}kC_{2n}^k=n2^{2n-1}$
\item $\displaystyle \sum_{k=1}^{n}(-1)^{k+1}\frac{1}{k}C_n^k[1-(1-x)^k]=\sum_{k=1}^{n}\frac{x^k}{k}$
\item $\displaystyle \sum_{k=0}^{n}\frac{(-1)^k}{2k+1}C_n^k=\frac{2^{2n}}{2n+1}(C_{2n}^n)^{-1}$
\item $\displaystyle \sum_{k=0}^{n}\frac{(-1)^k}{1-2k}C_n^k=2^{2n}(C_{2n}^n)^{-1}$
\item $\displaystyle \sum_{k=0}^{n}kC_{2n+1}^k=(2n+1)2^{2n-1}-\frac{2n+1}{2}C_{2n}^n$
\item $\displaystyle \sum_{k=0}^{n}kC_{2n}^{n-k}=nC_{2n-1}^{n}$
\item $\displaystyle \sum_{k=0}^{n}kC_{2n+1}^{n-k}=(2n+1)C_{2n-1}^{n}-2^{2n-1}$
\item $\displaystyle \sum_{k=0}^{n}C_{2n+1}^{2k+1}x^k=2x^{-\frac{1}{2}}[(1+\sqrt{x})^{2n+1}-(1-\sqrt{x})^{2n+1}]$%30
\item $\displaystyle \sum_{k=0}^{[\frac{n}{2}]}C_{n-k}^k=
\frac{1}{\sqrt{5}}\left\{\left(\frac{1+\sqrt{5}}{2}\right)^{n+1}-\left(\frac{1-\sqrt{5}}{2}\right)^{n+1}\right\}$
\item $\displaystyle \sum_{k=0}^{[\frac{n}{2}]}C_{n-k}^kr^k=
\frac{1}{2^{n+1}\sqrt{1+4r}}\left\{(1+\sqrt{1+4r})^{n+1}-(1-\sqrt{1+4r})^{n+1}\right\}$
\item $\displaystyle \sum_{k=0}^{[\frac{n}{2}]}C_{n-k}^k2^k=\frac{1}{3}(2^{n+1}+(-1)^n)$
\item $\displaystyle \sum_{k=0}^{[\frac{n}{2}]}C_{n-k}^k6^k=\frac{1}{5}(3^{n+1}+(-1)^n2^{n+1})$
\item $\displaystyle \sum_{k=0}^{[\frac{n}{2}]}(-1)^kC_{n-k}^k=\frac{2}{\sqrt{3}}\sin\frac{n+1}{3}\pi$
\item $\displaystyle \sum_{k=0}^{n-1}C_{n+k}^{2k+1}=
\frac{1}{\sqrt{5}}\left\{\left(\frac{3+\sqrt{5}}{2}\right)^{n}-\left(\frac{3-\sqrt{5}}{2}\right)^{n}\right\}$
\item $\displaystyle \sum_{k=0}^{n}C_{n+k}^{2k}=
\frac{1}{\sqrt{5}}\left\{\left(\frac{1+\sqrt{5}}{2}\right)^{2n+1}-\left(\frac{1-\sqrt{5}}{2}\right)^{2n+1}\right\}$
\item $\displaystyle \sum_{k=0}^{m}C_{k+n-1}^{n-1}=C_{m+n}^{n}$
\item $\displaystyle \sum_{k=0}^{n}2^{n-k}C_{n+k}^{2k}=\frac{1}{3}(2^{2n+1}+1)$
\item $\displaystyle \sum_{k=0}^{n}(-1)^kC_n^k\left(1+\frac{1}{2}+\cdots+\frac{1}{k}\right)=-\frac{1}{n}$%40
\item $\displaystyle \sum_{k=0}^{n}(-1)^kC_n^k\left(x+\frac{x^2}{2}+\cdots+\frac{x^k}{k}\right)=-\frac{1}{n}[1-(1-x)^n]$
\item $\displaystyle \sum_{k=0}^{n}C_{n+p}^{k+p}x^k=\sum_{k=0}^{n}C_{p+k-1}^n(1+x)^{n-k}$
\item $\displaystyle \sum_{k=0}^{n}(-1)^kC_{n+p}^{k+p}=C_{n+p-1}^n$
\item $\displaystyle \sum_{k=0}^{[\frac{n}{2}]}(-1)^k\frac{n}{n-k}C_{n-k}^kx^{n-2k}=
\frac{1}{2^n}\left\{(x+\sqrt{x^2-4})^n+(x-\sqrt{x^2-4})^n\right\}$
\item $\displaystyle \sum_{k=0}^{[\frac{n}{2}]}(-1)^k\frac{n}{n-k}C_{n-k}^k2^{n-2k}=2$
\item $\displaystyle \sum_{k=0}^{[\frac{n}{2}]}(-1)^k\frac{n}{n-k}C_{n-k}^k=2\cos\frac{n\pi}{3},n=1,2,\cdots.$
\item $\displaystyle \sum_{k=0}^{[\frac{n}{2}]}C_n^k\cos\frac{(n-2k)}{3}\pi
=\left\{ {\begin{array}{ll}
\displaystyle\frac{1}{2},&n\text{为奇数}\\[1.5ex]
\displaystyle\frac{1}{2}(1+C_n^{\frac{n}{2}}),&n\text{为偶数}
\end{array}} \right.$
\item $\displaystyle \sum_{k=0}^{[\frac{n}{2}]}\frac{(n+1-2k)^2}{n+1-k}C_n^k=2^n$
\item $\displaystyle \sum_{k=0}^{n}\frac{(-1)^k}{k+1}C_n^k[(1+x)^{k+1}-1]=(-1)^n\frac{x^{n+1}}{n+1}$
\item $\displaystyle \sum_{k=0}^{n}\frac{(-1)^k}{k+1}C_n^k\left(1+\frac{1}{2}+\cdots+\frac{1}{k+1}\right)=\frac{1}{(n+1)^2}$%50
\item $\displaystyle \sum_{k=1}^{n}(-1)^{k+1}C_n^k\sum_{l=1}^k\frac{1}{l}\left(1+\frac{1}{2}+\cdots+\frac{1}{l}\right)=\frac{1}{n^2}$
\item $\displaystyle \sum_{k=0}^{n}\frac{(-1)^{k}}{k+1}C_n^k\sum_{l=0}^k\frac{1}{l+1}\left(1+\frac{1}{2}+\cdots+\frac{1}{l+1}\right)
    =\frac{1}{(n+1)^3}$
\item $\displaystyle \sum_{k=0}^{[\frac{n}{2}]}(-1)^kC_{n-k}^k2^{n-2k}=n+1$
\item $\displaystyle \sum_{k=0}^{[\frac{n}{2}]}(-1)^kC_{n-k}^k\frac{(2\cos x)^{n-2k}}{n-k}=\frac{2}{n}\cos nx$
\item $\displaystyle \sum_{k=0}^{[\frac{n}{2}]}C_n^k\cos(n-2k)x
=\left\{ {\begin{array}{ll}
\displaystyle 2^{n-1}\cos^nx,&n\text{为奇数}\\[1.5ex]
\displaystyle 2^{n-1}\cos^nx+\frac{1}{2}C_n^{\frac{n}{2}},&n\text{为偶数}
\end{array}} \right.$
\item $\displaystyle \sum_{k=1}^{n}\frac{(-1)^{k+1}}{k}C_n^k\left(1+\frac{1}{2}+\cdots+\frac{1}{k}\right)=\sum_{k=1}^n\frac{1}{k^2}$
\item $\displaystyle \sum_{k=1}^{n}(-1)^{k}C_n^k\left(1+\frac{1}{2^2}+\cdots+\frac{1}{k^2}\right)=-\frac{1}{n}\sum_{k=1}^n\frac{1}{k}$
\item $\displaystyle \sum_{k=0}^{n}(-1)^kC_n^k(x+n-k)^n=n!$
\item $\displaystyle \sum_{k=0}^{n}(-1)^kP(k)C_n^k=0$这里$P(k)$是$k$的$p(<n)$次多项式
\item $\displaystyle \sum_{k=0}^{n}(-1)^kC_n^k(x-k)^{n+1}=\left(x-\frac{n}{2}\right)(n+1)!$%60
\item $\displaystyle \sum_{k=1}^{n}(-1)^kC_n^k\left(1+\frac{1}{3}+\cdots+\frac{1}{2k-1}\right)=-\frac{2^{2n-2}}{2n-1}(C_{2n-2}^{n-1})^{-1}$
\item $\displaystyle \sum_{k=1}^{n}k^4C_n^k=C_n^12^{n-1}+14C_n^22^{n-2}+36C_n^32^{n-3}+24C_n^42^{n-4}$
\item $\displaystyle \sum_{k=0}^{n}(-1)^k(k+1)^nC_{n+1}^{k+1}=0$
\item $\displaystyle \sum_{k=0}^{n}\sum_{l=0}^{k}(-1)^lC_k^l(l+1)^n=0$\\
在65-73的等式中,$\{a_n\}$是$p$阶等差数列,$\Delta^ka_0$是$\{a_n\}$的$k$阶差分数列首项,$(n\geqslant p)$
\item $\displaystyle \sum_{k=0}^{n}a_kC_n^k=\sum_{k=0}^{p}2^{n-k}C_n^k\Delta^ka_0$
\item $\displaystyle \sum_{k=0}^{n}(-1)^ka_kC_n^k=0(n>p)$
\item $\displaystyle \sum_{k=0}^{n}a_kC_n^kx^k(1-x)^{n-k}=\sum_{k=0}^{p}C_n^k\Delta^ka_0x^k$
\item $\displaystyle \sum_{k=0}^{n}a_kC_n^kt^k=\sum_{k=0}^{p}C_n^kt^k(1+t)^{n-k}\Delta^ka_0$
\item $\displaystyle \sum_{k=0}^{n}(-1)^ka_kC_n^kt^k=\sum_{k=0}^{p}(-1)^kC_n^kt^k(1-t)^{n-k}\Delta^ka_0$
\item $\displaystyle \sum_{k=0}^{n}a_kC_n^k\cos kx=
\sum_{k=0}^{p}C_n^k2^{n-k}\left(\cos \frac{x}{2}\right)^{n-k}\cos\frac{n+k}{2}x\Delta^ka_0$%70
\item $\displaystyle \sum_{k=0}^{n}a_kC_n^k\sin kx=
\sum_{k=0}^{p}C_n^k2^{n-k}\left(\cos \frac{x}{2}\right)^{n-k}\sin\frac{n+k}{2}x\Delta^ka_0$
\item $\displaystyle \sum_{k=0}^{n}(-1)^ka_kC_n^k\cos kx=
(-1)^n\sum_{k=0}^{p}C_n^k2^{n-k}\cos \frac{1}{2}[n(x+\pi)+k(x-\pi)]\left(\sin \frac{x}{2}\right)^{n-k}\Delta^ka_0$
\item $\displaystyle \sum_{k=0}^{n}(-1)^ka_kC_n^k\sin kx=
(-1)^n\sum_{k=0}^{p}C_n^k2^{n-k}\sin \frac{1}{2}[n(x+\pi)+k(x-\pi)]\left(\sin \frac{x}{2}\right)^{n-k}\Delta^ka_0$
\item $\displaystyle \sum_{k=0}^{n}C_n^k\cos kx=2^n\left(\cos\frac{x}{2}\right)^n\cos\frac{x}{2}x$
\item $\displaystyle \sum_{k=0}^{n}C_n^k\sin kx=2^n\left(\cos\frac{x}{2}\right)^n\sin\frac{x}{2}x$
\item $\displaystyle \sum_{k=0}^{n}(-1)^kC_n^k\cos kx=(-1)^n2^n\cos\frac{n}{2}(x+\pi)\left(\sin\frac{x}{2}\right)^n$
\item $\displaystyle \sum_{k=0}^{n}(-1)^kC_n^k\sin kx=(-1)^n2^n\sin\frac{n}{2}(x+\pi)\left(\sin\frac{x}{2}\right)^n$
\item $\displaystyle \sum_{k=0}^{n}kC_n^k\cos kx=n2^{n-1}\left(\cos\frac{x}{2}\right)^{n-1}\cos\frac{n+1}{2}x$
\item $\displaystyle \sum_{k=0}^{n}kC_n^k\sin kx=n2^{n-1}\left(\cos\frac{x}{2}\right)^{n-1}\sin\frac{n+1}{2}x$
\item $\displaystyle \sum_{k=0}^{n}(-1)^kkC_n^k\cos kx=(-1)^nn2^{n-1}\cos \frac{1}{2}[(n+1)x+(n-1)\pi]\left(\sin \frac{x}{2}\right)^{n-1}$%80
\item $\displaystyle \sum_{k=0}^{n}(-1)^kkC_n^k\sin kx=(-1)^nn2^{n-1}\sin \frac{1}{2}[(n+1)x+(n-1)\pi]\left(\sin \frac{x}{2}\right)^{n-1}$
\item $\displaystyle \sum_{k=0}^{n}(-1)^kC_n^k2^k\left(\cos\frac{x}{2}\right)^k\cos\frac{k}{2}x=(-1)^n\cos nx$
\item $\displaystyle \sum_{k=0}^{n}(-1)^kC_n^k2^k\left(\cos\frac{x}{2}\right)^k\sin\frac{k}{2}x=(-1)^n\sin nx$
\item $\displaystyle \sum_{k=0}^{n}C_n^k2^k\cos\frac{k}{2}(x+\pi)\left(\sin\frac{x}{2}\right)^k=\cos nx$
\item $\displaystyle \sum_{k=0}^{n}C_n^k2^k\sin\frac{k}{2}(x+\pi)\left(\sin\frac{x}{2}\right)^k=\sin nx$
\item $\displaystyle \sum_{k=0}^{n}(-1)^k2^{\frac{k}{2}}C_n^k\cos\frac{\pi}{4}=\cos\frac{n\pi}{2}$
\item $\displaystyle \sum_{k=0}^{n}(-1)^{k+1}2^{\frac{k}{2}}C_n^k\sin\frac{\pi}{4}=\sin\frac{n\pi}{2}$
\item $\displaystyle \sum_{k=0}^{[\frac{n-r}{m}]}C_n^{r+km}=
    \frac{1}{m}\sum_{k=0}^{m-1}\left(2\cos\frac{k\pi}{m}\right)\cos\frac{(n-2r)k\pi}{m},r\leqslant m$
\item $\displaystyle \sum_{k=0}^{[\frac{n}{2}]}C_n^{2k}=2^{n-1}$
\item $\displaystyle \sum_{k=0}^{[\frac{n}{4}]}C_n^{4k}=\frac{1}{4}\left(2^n+2(\sqrt{n})^n\cos\frac{n\pi}{4}\right)$%90
\item $\displaystyle \sum_{k=0}^{n}C_{4n}^{4k}=\frac{1}{4}(2^{4n}+(-1)^n2^{2n+1})$
\item $\displaystyle \sum_{k=0}^{[\frac{n-1}{3}]}C_n^{1+3k}=\frac{1}{3}\left(2^n+2\cos\frac{(n-2)}{2}\pi\right)$
\item $\displaystyle \sum_{k=0}^{[\frac{n-2}{3}]}C_n^{2+3k}=\frac{1}{3}\left(2^n+2\cos\frac{(n-4)}{2}\pi\right)$
\item $\displaystyle \sum_{k=0}^{n-1}(-1)^k\left(\cos\frac{k\pi}{n}\right)^n=\frac{n}{2^{n-1}}$
\item $\displaystyle \sum_{k=0}^{[\frac{n-r}{m}]}C_n^{r+km}\cos(r+km)x
=\frac{2^n}{m}\sum_{k=0}^{m-1}\cos\left[\frac{nx}{2}+\frac{k\pi}{m}(n-2r)\right]\cos^n\left(\frac{x}{2}+\frac{k\pi}{m}\right)$
\item $\displaystyle \sum_{k=0}^{[\frac{n-r}{m}]}C_n^{r+km}\sin(r+km)x
=\frac{2^n}{m}\sum_{k=0}^{m-1}\sin\left[\frac{nx}{2}+\frac{k\pi}{m}(n-2r)\right]\cos^n\left(\frac{x}{2}+\frac{k\pi}{m}\right)$
\item $\displaystyle \sum_{k=0}^{[\frac{n}{m}]}C_n^{km}\cos kmx
=\frac{2^n}{m}\sum_{k=0}^{m-1}\cos n\left(\frac{x}{2}+\frac{k\pi}{m}\right)\cos^n\left(\frac{x}{2}+\frac{k\pi}{m}\right)$
\item $\displaystyle \sum_{k=0}^{[\frac{n}{m}]}C_n^{km}\sin kmx
=\frac{2^n}{m}\sum_{k=0}^{m-1}\sin n\left(\frac{x}{2}+\frac{k\pi}{m}\right)\cos^n\left(\frac{x}{2}+\frac{k\pi}{m}\right)$
\item $\displaystyle \sum_{k=0}^{[\frac{n}{2}]}C_n^{2k}\cos 2k=2^{n-1}
\left\{\cos\frac{nx}{2}\cos^n{\frac{x}{2}}+\cos\frac{x+\pi}{2}\left(\cos\frac{x+\pi}{2}\right)^n\right\}$
\item $\displaystyle \sum_{k=0}^{[\frac{n}{2}]}C_n^{2k}\sin 2k=2^{n-1}
\left\{\sin\frac{nx}{2}\cos^n{\frac{x}{2}}+\sin\frac{x+\pi}{2}\left(\cos\frac{x+\pi}{2}\right)^n\right\}$%100
\item $\displaystyle \sum_{k=0}^n\left(\frac{k}{n}-a\right)^2C_n^kx^k(1-x)^{n-k}=(x-a)^2+\frac{x(1-x)}{n}$
\item $\displaystyle \sum_{k=0}^{[\frac{n}{2}]}(-1)^kC_{n-k}^k(2\cos x)^{n-2k}=\frac{\sin(n+1)x}{\sin x}$
\item $\displaystyle \sum_{k=0}^n(-1)^kk2^{k-1}C_n^k\left(\cos\frac{x}{2}\right)^{k-1}\cos\frac{k+1}{2}x=(-1)^nn\cos nx$
\item $\displaystyle \sum_{k=0}^n(-1)^kk2^{k-1}C_n^k\left(\cos\frac{x}{2}\right)^{k-1}\sin\frac{k+1}{2}x=(-1)^nn\sin nx$
\item $\displaystyle \sum_{k=0}^nk2^{k-1}C_n^k\cos\frac{1}{2}[(k+1)x+(k-1)\pi]\left(\sin\frac{x}{2}\right)^{k-1}=n\cos nx$
\item $\displaystyle \sum_{k=0}^nk2^{k-1}C_n^k\sin\frac{1}{2}[(k+1)x+(k-1)\pi]\left(\sin\frac{x}{2}\right)^{k-1}=n\sin nx$
\item $\displaystyle \sum_{k=0}^n(-1)^kk2^{k-1}C_n^k2^{\frac{k-1}{2}}\sin\frac{k+1}{4}\pi=(-1)^nn\sin\frac{n\pi}{2}$
\item $\displaystyle \sum_{k=0}^{[\frac{n-1}{2}]}C_n^{2k+1}\cos (2k+1)x=2^{n-1}\left\{\cos^n\frac{x}{2}\cos\frac{nx}{2}
-(-1)^n\sin^n\frac{x}{2}\cos\frac{n(x+\pi)}{2}\right\}$
\item $\displaystyle \sum_{k=0}^{[\frac{n-1}{2}]}C_n^{2k+1}\sin (2k+1)x=2^{n-1}\left\{\cos^n\frac{x}{2}\sin\frac{nx}{2}
-(-1)^n\sin^n\frac{x}{2}\sin\frac{n(x+\pi)}{2}\right\}$
\item $\displaystyle \sum_{k=0}^{n}C_n^k(x+k)^{k-1}(y+n-k)^{n-k}=x^{-1}(x+y+n)^n$%110
\item $\displaystyle \sum_{k=0}^{n}C_n^k(x+k)^{k-1}(y+n-k)^{n-k-1}=(x^{-1}+y^{-1})(x+y+n)^{n-1}$
\item $\displaystyle \sum_{k=0}^{n}C_n^k(x+k)^{k-2}(y+n-k)^{n-k}=x^{-2}(x+1)^{-1}[(x+1)(x+y+n)^n-nx(x+y+n)^{n-1}]$
\item $\displaystyle \sum_{k=0}^{n}C_n^k(x+k)^k(y+n-k)^{n-k}=\sum_{k=0}^{n}C_n^kk!(x+y+n)^{n-k}$
\item $\displaystyle \sum_{k=1}^{n}C_{n-1}^{k-1}n^{-k}k!=1$
\item $\displaystyle \sum_{k=0}^{n}C_n^k(x+k)(x+n)^{n-k-1}k!=(x+n)^n$
\item $\displaystyle \sum_{k=0}^{n}C_n^kx(x+k)^{k-1}(y-k)^{n-k}=(x+y)^n$
\item $\displaystyle \sum_{k=0}^{n}C_n^k(x-k)^{n-k}y(y+k)^{k-1}=(x+y)^n$
\item $\displaystyle \sum_{k=0}^{n}C_n^k(x+k)^k(y+n-k)^{n-k-1}=y^{-1}(x+y+n)^n$
\item $\displaystyle \sum_{k=0}^{n}C_n^kk^k(n-k+1)^{n-k-1}=(n+1)^n$
\item $\displaystyle \sum_{k=0}^{n}(-1)^kC_n^k(n-k-1)^{n-k-1}(k+1)^k=-n^n$%120
\item $\displaystyle \sum_{k=0}^{n}C_n^k(x^2-k)(-x+k)^{k-2}(x+y+n-k)^{n-k}=y(y+n)^{n-1}$
\item $\displaystyle \sum_{k=1}^{n}C_n^kk^{k-1}(n-k+1)^{n-k}=n(n+1)^{n-1}$
\item $\displaystyle \sum_{k=m}^{n}(-1)^kC_n^kC_k^m=(-1)^m\delta_{mn}$
\item $\displaystyle \sum_{k=m}^{n}C_n^kC_k^m=2^{n-m}C_n^m$
\item $\displaystyle \sum_{k=1}^{n-1}\frac{1}{k(n-k)}C_{2(k-1)}^{k-1}C_{2(n-k-1)}^{n-k-1}=\frac{1}{n}C_{2(n-1)}^{n-1}$
\item $\displaystyle \sum_{k=l}^{n}(-1)^k\frac{1}{k+1}C_n^kC_k^l=\frac{(-1)^l}{n+1}$
\item $\displaystyle \sum_{k=m}^{n}C_n^kC_k^mx^k(1-x)^{n-k}=C_n^mx^m$
\item $\displaystyle \sum_{k=0}^{n}(C_n^k)^2=C_{2n}^n$
\item $\displaystyle \sum_{k=0}^{n}(-1)^k(C_n^k)^2=
\left\{ {\begin{array}{ll}
\displaystyle 0,&n\text{为奇数}\\[1.5ex]
\displaystyle(-1)^\frac{n}{2}C_n^{\frac{n}{2}},&n\text{为偶数}
\end{array}} \right.$
\item $\displaystyle \sum_{k=0}^{n}C_{p+k}^pC_{q+n-k}^q=C_{p+q+1}^{p+q+n+1}$%130
\item $\displaystyle \sum_{k=0}^{q}C_n^kC_m^{q-k}=C_{m+n}^q$
\item $\displaystyle \sum_{k=0}^{n}(-1)^kC_n^kC_{m+k}^q=(-1)^nC_m^{q-n}$
\item $\displaystyle \sum_{k=0}^{n}C_{2k}^kC_{2n-2k}^{n-k}=2^{2n}$
\item $\displaystyle \sum_{k=0}^{n-1}(C_n^0+C_n^1+\cdots+C_n^k)(C_n^{k+1}+C_n^{k+2}+\cdots+C_n^n)=\frac{n}{2}C_{2n}^n$
\item $\displaystyle \sum_{k=1}^{n}C_n^{k-1}C_n^k=\frac{(2n)!}{(n-1)!(n+1)!}$
\item $\displaystyle \sum_{k=0}^{2n}C_m^kC_m^{2n-k}=(-1)^nC_m^n$
\item $\displaystyle \sum_{k=0}^{n}kC_n^kC_m^k=nC_{n+m-1}^n$
\item $\displaystyle \sum_{k=0}^{n}(-1)^kC_n^kC_{m+k}^{q+k}=(-1)^nC_m^{q+n}$
\item $\displaystyle \sum_{k=0}^{n}kC_{2k}^kC_{2n-2k}^{n-k}=n2^{2n-1}$
\item $\displaystyle \sum_{k=0}^{n}(-1)^kC_n^kC_{m+n+k}^m=\delta_{mn}(m\leqslant n)$%140
\item $\displaystyle \sum_{k=0}^{n-1}(C_{2n}^{2k+1})^2=\frac{1}{2}(C_{4n}^{2n}+(-1)^{n-1}C_{2n}^n)$
\item $\displaystyle \sum_{k=0}^{[\frac{n}{2}]}(-1)^kC_{n+1}^kC_{2n-2k}^n=n+1$
\item $\displaystyle \sum_{k=0}^{[\frac{n}{2}]}(-1)^kC_n^kC_{2n-2k}^n=2^n$
\item $\displaystyle \sum_{k=0}^{[\frac{n-1}{2}]}\left(1-\frac{2k}{n}\right)^2(C_n^k)^2=\frac{1}{n}C_{2n-1}^{n-1}$
\item $\displaystyle \sum_{k=q}^{r}(-1)^kC_k^qC_n^{r-k}=
\left\{ {\begin{array}{ll}
\displaystyle (-1)^qC_{n-q-1}^{r-q},&r<n\\[1.5ex]
\displaystyle 0,&r\geqslant n
\end{array}} \right.$
\item $\displaystyle \sum_{k=0}^{2n}(-1)^kC_{2k}^kC_{4n-2k}^{2n-k}=2^{2n}C_{2n}^n$
\item $\displaystyle \sum_{k=0}^{2n+1}(-1)^kC_{2k}^kC_{4n+2-2k}^{2n+1-k}=0$
\item $\displaystyle \sum_{k=0}^{n}C_{2n-2k}^{n-k}C_{2k}^k\frac{1}{2k-1}=0$
\item $\displaystyle \sum_{k=0}^{n-1}(C_n^0-C_n^1+C_n^2-\cdots+(-1)^kC_n^k)
((-1)^{k+1}C_n^{k+1}+(-1)^{k+2}C_n^{k+2}+\cdots+(-1)^nC_n^n)=-C_{2n-2}^{n-1}$
\item $\displaystyle \sum_{k=0}^{n}(-1)^kC_{n+p}^{k+p}C_{p+k-1}^k=1$%150
\item $\displaystyle \sum_{k=0}^{n}(-1)^kkC_n^kC_{n+m-1}^k=(-1)^nnC_m^n$
\item $\displaystyle \sum_{k=0}^{n}(-1)^kC_{p_1+k}^{p_1}C_{p_2+k}^{p_2}\cdots C_{p_r+k}^{p_r}C_n^k=0,
n>p_1+p_2+\cdots+p_r$
\item $\displaystyle \sum_{k=0}^{n}(-1)^k(C_{p+k}^p)^rC_n^k=0,n>rp$
\item $\displaystyle \sum_{k=0}^{n}(-1)^kC_n^kC_{m-2k}^{n-1}=0,m\geqslant 2n$
\item $\displaystyle \sum_{k=0}^{n}(-1)^k(C_n^k)^{-1}=\frac{n+1}{n+2}(1+(-1)^n)$
\item $\displaystyle \sum_{k=0}^{n}(-1)^k\frac{n-k}{C_{2n}^k}=0$
\item $\displaystyle \sum_{k=0}^{n}(-1)^kC_n^k(C_{m+k}^k)^{-1}=\frac{m}{m+n}$
\item $\displaystyle \sum_{k=0}^{n}(-1)^kC_n^k2^{2k}(C_{2k}^k)^{-1}=\frac{1}{1-2n}$
\item $\displaystyle \sum_{k=0}^{n}\frac{(-1)^k}{2k+1}C_n^k(C_{2k}^k)^{-1}2^{2k}=\frac{1}{2n+1}$
\item $\displaystyle \sum_{k=0}^{n}(-1)^{k-1}\frac{2^{2k}}{2k-1}C_n^k(C_{2k-2}^{k-1})^{-1}=
4\left(1+\frac{1}{3}+\cdots+\frac{1}{2n-1}\right)$%160

终于完结了

11# Tesla35

辛苦晒，加个威望以表谢意