Negative binomial theorem and redundant combinations

• $(1-x{)}^{-d}={\sum }_{n=0}^{\infty }{H}_{d-1}^{n+1}{x}^n$

• negative binomial theorem :

• $(1-x{)}^{-d}={\sum }_{n=0}^{\infty }(\genfrac{}{}{0ex}{}{n+d-1}{d-1})x^n$

• nested combination :

• $H_k^n=\left(\genfrac{}{}{0ex}{}{n+k-1}{k}\right)=\frac{(n+k-1)!}{k!(n-1)!}$

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