2項係数の総和
\(n\in\mathbb{N}_{0},m\in\mathbb{N}\)とする。
(1)
\[ \sum_{k=0}^{n}C(n,k)=2^{n} \]
(2)
\[ \sum_{k=0}^{n}P(k,m)C(n,k)=P(n,m)2^{n-m} \]
(3)
\[ \sum_{k=0}^{n}k^{m}C(n,k)=n\sum_{k=0}^{n-1}(k+1)^{m-1}C(n-1,k) \]
(4)
\[ \sum_{k=0}^{n}k^{m}C(n,k)=\sum_{j=0}^{m}S_{2}(m,j)P(n,j)2^{n-j} \]
(5)
\[ \sum_{k=0}^{n}kC(n,k)=n2^{n-1} \]
(6)
\[ \sum_{k=0}^{n}k^{2}C(n,k)=(n^{2}+n)2^{n-2} \]
(1)
\begin{align*} \sum_{k=0}^{n}C(n,k) & =\sum_{k=0}^{n}C(n,k)1^{k}1^{n-k}\\ & =(1+1)^{n}\\ & =2^{n} \end{align*}
(2)
\begin{align*} \sum_{k=0}^{n}P(k,m)C(n,k) & =\sum_{k=0}^{n}\frac{n!}{(k-m)!(n-k)!}\\ & =\frac{n!}{(n-m)!}\sum_{k=0}^{n}\frac{(n-m)!}{(k-m)!(n-k)!}\\ & =P(n,m)\sum_{k=0}^{n}C(n-m,n-k)\\ & =P(n,m)\sum_{k=0}^{n-m}C(n-m,k)\\ & =P(n,m)2^{n-m} \end{align*}
(3)
\begin{align*} \sum_{k=0}^{n}k^{m}C(n,k) & =n\sum_{k=0}^{n}k^{m-1}C(n-1,k-1)\\ & =n\sum_{k=0}^{n-1}(k+1)^{m-1}C(n-1,k) \end{align*}
(4)
\begin{align*} \sum_{k=0}^{n}k^{m}C(n,k) & =\sum_{k=0}^{n}\sum_{j=0}^{m}S_{2}(m,j)P(k,j)C(n,k)\\ & =\sum_{j=0}^{m}S_{2}(m,j)P(n,j)2^{n-j} \end{align*}
(5)
\begin{align*} \sum_{k=0}^{n}kC(n,k) & =\sum_{j=0}^{1}S_{2}(1,j)P(n,j)2^{n-j}\\ & =S_{2}(1,0)P(n,0)2^{n-0}+S_{2}(1,1)P(n,1)2^{n-1}\\ & =n2^{n-1} \end{align*}
(5)-2
\begin{align*} \sum_{k=0}^{n}kC(n,k) & =\sum_{k=0}^{n}nC(n-1,k-1)\\ & =n\sum_{k=1}^{n}C(n-1,k-1)\\ & =n\sum_{k=0}^{n-1}C(n-1,k)\\ & =n2^{n-1} \end{align*}
(5)-3
\begin{align*} \sum_{k=0}^{n}kC(n,k) & =\left[\frac{d}{dx}\sum_{k=0}^{n}C(n,k)x^{k}\right]_{x=1}\\ & =\left[\frac{d}{dx}(1+x)^{n}\right]_{x=1}\\ & =\left[n(1+x)^{n-1}\right]_{x=1}\\ & =n2^{n-1} \end{align*}
(6)
\begin{align*} \sum_{k=0}^{n}k^{2}C(n,k) & =\sum_{j=0}^{2}S_{2}(2,j)P(n,j)2^{n-j}\\ & =S_{2}(2,0)P(n,0)2^{n-0}+S_{2}(2,1)P(n,1)2^{n-1}+S_{2}(2,2)P(n,2)2^{n-2}\\ & =n2^{n-1}+n(n-1)2^{n-2}\\ & =(n^{2}+n)2^{n-2} \end{align*}
(6)-2
\begin{align*} \sum_{k=0}^{n}k^{2}C(n,k) & =\sum_{k=0}^{n}\left\{ k(k-1)C(n,k)+kC(n,k)\right\} \\ & =\sum_{k=0}^{n}\left\{ n(n-1)C(n-2,k-2)+nC(n-1,k-1)\right\} \\ & =n(n-1)\sum_{k=2}^{n}C(n-2,k-2)+n\sum_{k=1}^{n}C(n-1,k-1)\\ & =n(n-1)\sum_{k=0}^{n-2}C(n-2,k-2)+n\sum_{k=0}^{n-1}C(n-1,k-1)\\ & =n(n-1)2^{n-2}+n2^{n-1}\\ & =(n^{2}+n)2^{n-2} \end{align*}
(6)-3
\begin{align*} \sum_{k=0}^{n}k^{2}C(n,k) & =\left[\frac{d}{dx}x\frac{d}{dx}\sum_{k=0}^{n}C(n,k)x^{k}\right]_{x=1}\\ & =\left[\frac{d}{dx}x\frac{d}{dx}(1+x)^{n}\right]_{x=1}\\ & =n\left[\frac{d}{dx}x(1+x)^{n-1}\right]_{x=1}\\ & =n\left[\frac{d}{dx}\left\{ (1+x)^{n}-(1+x)^{n-1}\right\} \right]_{x=1}\\ & =n\left[n(1+x)^{n-1}-(n-1)(1+x)^{n-2}\right]_{x=1}\\ & =n\left\{ n2^{n-1}-(n-1)2^{n-2}\right\} \\ & =(n^{2}+n)2^{n-2} \end{align*}
ページ情報
タイトル | 2項係数の総和 |
URL | https://www.nomuramath.com/io7dswnn/ |
SNSボタン |