Lower fixed binomial coefficient → negative binomial theorem

$ \binom{A+i}{A} = [x^i] \sum_n \binom{A+n}{A}x^n = [x^i]\frac{1}{(1-x)^{A+1}}

$ (1-x)^{-d} = \sum_{n=0}^\infty \binom{n+d-1}{d-1}x^n

facing the opposite direction

$ [x^B]\frac{1}{(1-x)^{A}} = \binom{A+B-1}{A-1}

relevance

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