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*}
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2項係数の総和その他
\[
\sum_{k=1}^{n-1}\frac{C\left(k-n,k\right)}{k}=-H_{n-1}
\]
2項係数の関係その他
\[
C\left(\alpha,\beta\right)C\left(\beta,\gamma\right)=C\left(\alpha,\gamma\right)C\left(\alpha-\gamma,\beta-\gamma\right)
\]
一般ヴァンデルモンドの畳み込み定理
\[
\sum_{k_{1}+\cdots+k_{p}=m}\prod_{j=1}^{p}C\left(n_{j},k_{j}\right)=C\left(\sum_{j=1}^{p}n_{j},m\right)
\]
2項係数の第1引数と第2引数同士の総和
\[
\sum_{j=0}^{k-a}\left(-1\right)^{j}C\left(k,j+a\right)C\left(j+b,c\right)=\begin{cases}
\left(-1\right)^{k-a}C\left(b-a,c-k\right) & a-b+c\leq k\\
0 & k<a-b+c
\end{cases}
\]