パスカルの法則の応用
パスカルの法則の応用
\(n\in\mathbb{N}_{0}\)とする。
\(n\in\mathbb{N}_{0}\)とする。
(1)
\[ C\left(x+n,y+n\right)=C\left(x,y+n\right)+\sum_{k=0}^{n-1}C\left(x+k,y+n-1\right) \](2)
\[ C\left(x+n,y+n\right)=C\left(x,y\right)+\sum_{k=0}^{n-1}C\left(x+k,y+k+1\right) \](3)
\[ C\left(x+n,y+n\right)=\left(-1\right)^{n}\left\{ C\left(x+n,y\right)-\sum_{k=0}^{n-1}\left(-1\right)^{k}C\left(x+n+1,y+1+k\right)\right\} \](1)
\begin{align*} C\left(x+n,y+n\right) & =C\left(x+n-1,y+n\right)+C\left(x+n-1,y+n-1\right)\\ & =C\left(x,y+n\right)+\sum_{k=0}^{n-1}\left\{ C\left(x+n-k,y+n\right)-C\left(x+n-\left(k+1\right),y+n\right)\right\} \\ & =C\left(x,y+n\right)+\sum_{k=0}^{n-1}C\left(x+n-\left(k+1\right),y+n-1\right)\\ & =C\left(x,y+n\right)+\sum_{k=1}^{n}C\left(x+n-k,y+n-1\right)\\ & =C\left(x,y+n\right)+\sum_{k=0}^{n-1}C\left(x+k,y+n-1\right) \end{align*}(2)
\begin{align*} C\left(x+n,y+n\right) & =C\left(x+n-1,y+n\right)+C\left(x+n-1,y+n-1\right)\\ & =C\left(x,y\right)+\sum_{k=0}^{n-1}\left\{ C\left(x+n-k,y+n-k\right)-C\left(x+n-\left(k+1\right),y+n-\left(k+1\right)\right)\right\} \\ & =C\left(x,y\right)+\sum_{k=0}^{n-1}C\left(x+n-\left(k+1\right),y+n-k\right)\\ & =C\left(x,y\right)+\sum_{k=0}^{n-1}C\left(x+k,y+k+1\right) \end{align*}(3)
\begin{align*} C\left(x+n,y+n\right) & =-C\left(x+n,y+n-1\right)+C\left(x+n+1,y+n\right)\\ & =\left(-1\right)^{n}C\left(x+n,y\right)-\sum_{k=0}^{n-1}\left\{ \left(-1\right)^{k+1}C\left(x+n,y+n-k\right)-\left(-1\right)^{k}C\left(x+n,y+n-1-k\right)\right\} \\ & =\left(-1\right)^{n}C\left(x+n,y\right)-\sum_{k=0}^{n-1}\left(-1\right)^{k+1}\left\{ C\left(x+n,y+n-k\right)+C\left(x+n,y+n-1-k\right)\right\} \\ & =\left(-1\right)^{n}C\left(x+n,y\right)-\sum_{k=0}^{n-1}\left(-1\right)^{k+1}\left\{ C\left(x+n+1,y+n-k\right)\right\} \\ & =\left(-1\right)^{n}C\left(x+n,y\right)+\sum_{k=0}^{n-1}\left(-1\right)^{k}C\left(x+n+1,y+n-k\right)\\ & =\left(-1\right)^{n}C\left(x+n,y\right)+\sum_{k=0}^{n-1}\left(-1\right)^{n-1-k}C\left(x+n+1,y+1+k\right)\\ & =\left(-1\right)^{n}\left\{ C\left(x+n,y\right)-\sum_{k=0}^{n-1}\left(-1\right)^{k}C\left(x+n+1,y+1+k\right)\right\} \end{align*}ページ情報
| タイトル | パスカルの法則の応用 |
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2項係数の2乗和
\[
\sum_{j=0}^{m}C^{2}(m,j)=C(2m,m)
\]
2項係数の特殊な積
\[
C(x,t)C(t,y)=C(x,y)C(x-y,x-t)
\]
2項係数の相加平均・相乗平均を含む極限
\[
\lim_{n\rightarrow\infty}\sqrt[n]{\sqrt[n+1]{\prod_{k=0}^{n}C\left(n,k\right)}}=\sqrt{e}
\]
飛び飛びの2項定理
\[
\sum_{k=0}^{\infty}C\left(n,2k\right)a^{2k}b^{n-2k}=\frac{1}{2}\left\{ \left(a+b\right)^{n}+\left(-a+b\right)^{n}\right\}
\]