User talk:Baijiangchen

If:

$W(0):=1$

$W(n):=\sum_{i=0}^{n-1}({n-1 \choose i}W(i)(x-i-1)!(2^{x-i-1}))$

Then:

$W(n)=(2n-1)!!$

Sam's stuff

Let $W(n)=\sum_{k=1}^{n}(\binom{x-1}{k-1}W(k-1)(n-k)!(2^{n-k}))$

Assume that for some integer $x$, $W(x)=(2x-1)!!$. We intend to show that $W(x+1)=(2(x+1)-1)!!=(2x+1)!!$.

$W(x+1)=\sum_{k=1}^{x+1}(\binom{x}{k-1}W(k-1)(x-k+1)!(2^{x-k+1}))$

$=\sum_{k=1}^{x}(\binom{x-1}{k-1}(\frac{x}{x-k+1})W(k-1)(x-k)!(x-k+1)(2^{x-k})(2))+\binom{x}{x}W(x)(0)!(2^{0})$

$=2x\sum_{k=1}^{x}(\binom{x-1}{k-1}W(k-1)(x-k)!(2^{x-k}))+W(x)$

$=2x(2x-1)!!+(2x-1)!!=(2x-1)!!(2x+1)=(2x+1)!!$

Q.E.D.

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