How did they get the following:
$$\frac {x^n}{(1-x)^n} = \sum\limits_{m}{m-1 \choose n-1}{x^m}$$
2026-04-06 23:57:01.1775519821
Expressing $\frac {x^n}{(1-x)^n}$ as a generating function
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Since $\displaystyle \frac{1}{(1-x)^n}=(1-x)^{-n}=\sum_{k=0}^{\infty}\binom{n+k-1}{k}x^k$,
$\displaystyle \frac{x^n}{(1-x)^n}=\sum_{k=0}^{\infty}\binom{n+k-1}{k}x^{n+k}=\sum_{m=n}^{\infty}\binom{m-1}{m-n}x^m=\sum_{m=n}^{\infty}\binom{m-1}{n-1}x^m$ $\;\;$(letting $m=n+k$)