2項係数の1項間漸化式
2項係数は以下の1項間漸化式を満たす。
(1)
\[ C(x+1,y)=\frac{x+1}{x+1-y}C(x,y) \]
(2)
\[ C(x-1,y)=\frac{x-y}{x}C(x,y) \]
(3)
\[ C(x,y+1)=\frac{x-y}{y+1}C(x,y) \]
(4)
\[ C(x,y-1)=\frac{y}{x-y+1}C(x,y) \]
(5)
\[ C(x+1,y+1)=\frac{x+1}{y+1}C(x,y) \]
(6)
\[ C(x+1,y-1)=\frac{(x+1)y}{(x-y+1)(x-y+2)}C(x,y) \]
(7)
\[ C(x-1,y+1)=\frac{(x-y-1)(x-y)}{x(y+1)}C(x,y) \]
(8)
\[ C(x-1,y-1)=\frac{y}{x}C(x,y) \]
(1)
\begin{align*} C(x+1,y) & =\frac{(x+1)!}{y!(x+1-y)!}\\ & =\frac{x+1}{x+1-y}\frac{x!}{y!(x-y)!}\\ & =\frac{x+1}{x+1-y}C(x,y) \end{align*}
(2)
\begin{align*} C(x-1,y) & =\frac{(x-1)!}{y!(x-1-y)!}\\ & =\frac{x-y}{x}\frac{x!}{y!(x-y)!}\\ & =\frac{x-y}{x}C(x,y) \end{align*}
(3)
\begin{align*} C(x,y+1) & =\frac{x!}{(y+1)!(x-y-1)!}\\ & =\frac{x-y}{y+1}\frac{x!}{y!(x-y)!}\\ & =\frac{x-y}{y+1}C(x,y) \end{align*}
(4)
\begin{align*} C(x,y-1) & =\frac{x!}{(y-1)!(x-y+1)!}\\ & =\frac{y}{x-y+1}\frac{x!}{y!(x-y)!}\\ & =\frac{y}{x-y+1}C(x,y) \end{align*}
(5)
\begin{align*} C(x+1,y+1) & =\frac{(x+1)!}{(y+1)!(x-y)!}\\ & =\frac{x+1}{y+1}\frac{x!}{y!(x-y)!}\\ & =\frac{x+1}{y+1}C(x,y) \end{align*}
(6)
\begin{align*} C(x+1,y-1) & =\frac{(x+1)!}{(y-1)!(x-y+2)!}\\ & =\frac{(x+1)y}{(x-y+1)(x-y+2)}\frac{x!}{y!(x-y)!}\\ & =\frac{(x+1)y}{(x-y+1)(x-y+2)}C(x,y) \end{align*}
(7)
\begin{align*} C(x-1,y+1) & =\frac{(x-1)!}{(y+1)!(x-y-2)!}\\ & =\frac{(x-y-1)(x-y)}{x(y+1)}\frac{x!}{y!(x-y)!}\\ & =\frac{(x-y-1)(x-y)}{x(y+1)}C(x,y) \end{align*}
(8)
\begin{align*} C(x-1,y-1) & =\frac{(x-1)!}{(y-1)!(x-y)!}\\ & =\frac{y}{x}\frac{x!}{y!(x-y)!}\\ & =\frac{y}{x}C(x,y) \end{align*}
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