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Oct 30, 2021 at 21:39 comment added Noam D. Elkies Already for squares of trinomial coefficients there is a lot of literature but no closed formula (see oeis.org/A002893). For quadrinomials, see oeis.org/A002895 which also gives extensive links, and also the name "Domb numbers" [Named after the British-Israeli theoretical physicist Cyril Domb (1920-2012)"], but no closed form despite the appearance of $256 = 4^4$ as the $n=3$ sum.
Oct 30, 2021 at 2:04 history edited Sam Hopkins
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Oct 29, 2021 at 15:17 answer added T. Amdeberhan timeline score: 9
Oct 28, 2021 at 23:29 comment added Sam Hopkins See mathoverflow.net/questions/128249/…. It is unlikely there is any "nice" exact answer.
Oct 28, 2021 at 23:27 history edited Sam Hopkins CC BY-SA 4.0
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Oct 28, 2021 at 23:18 history asked Miklos Bona CC BY-SA 4.0