$\binom{A+i}{A} = [x^i] \sum_n \binom{A+n}{A}x^n = [x^i]\frac{1}{(1-x)^{A+1}} $
<ul>
<li><a href="/en/negative%20binomial%20theorem">negative binomial theorem</a>
</li>
<li>$(1-x)^{-d} = \sum_{n=0}^\infty \binom{n+d-1}{d-1}x^n$
</li>
<li><a href="/en/binomial%20coefficient%20of%20under-fixation">binomial coefficient of under-fixation</a> → <a href="/en/negative%20binomial%20theorem">negative binomial theorem</a>
</li>
</ul>
facing the opposite direction
<ul>
<li><a href="/en/Power%20series%20%E2%86%92%20binomial%20coefficient">Power series → binomial coefficient</a> :</li>
<li>$[x^B]\frac{1}{(1-x)^{A}} = \binom{A+B-1}{A-1}$</li>
</ul>
relevance
<ul>
<li><a href="https://scol.hatenablog.com/entry/2019/06/02/190000">Infinite sums with respect to the first argument of the binomial coefficients (mother function) - Skol&#39;s Wisdom Book</a><ul>
<li>This is <a href="/en/generating%20function">generating function</a>.</li>
</ul>
</li>
</ul>
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Lower fixed binomial coefficient → negative binomial theorem