What is the value or tight upper limit of the following summation:
$$\sum_{k=0}^n{n\choose k} x^{k(n-k)}$$
2026-04-04 02:35:11.1775270111
Sum of combinations series
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It is a polynomial in $x$ of degree $(n/2)^2$ if $n$ is even, $(n^2-1)/4$ if $n$ is odd. The largest coefficients are those for $k$ near $n/2$, and those are also the terms with highest degree. If you are interested in estimates that are tight for large $x$, that's the place to look...