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A331657 a(n) = Sum_{k=0..n} (-1)^(n - k) * binomial(n,k) * binomial(n+k,k) * n^k.

%I A331657 #26 Aug 30 2025 09:12:22
%S A331657 1,1,13,305,10321,458649,25289461,1666406209,127779121345,
%T A331657 11178899075537,1098961472475901,119937806278590321,
%U A331657 14389588419704763409,1882432013890951832425,266678501426944160023653,40673387011956179149166849,6644919093900517186643470081
%N A331657 a(n) = Sum_{k=0..n} (-1)^(n - k) * binomial(n,k) * binomial(n+k,k) * n^k.
%H A331657 Seiichi Manyama, <a href="/A331657/b331657.txt">Table of n, a(n) for n = 0..322</a>
%H A331657 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/LegendrePolynomial.html">Legendre Polynomial</a>
%F A331657 a(n) = central coefficient of (1 + (2*n - 1)*x + n*(n - 1)*x^2)^n.
%F A331657 a(n) = [x^n] 1 / sqrt(1 - 2*(2*n - 1)*x + x^2).
%F A331657 a(n) = n! * [x^n] exp((2*n - 1)*x) * BesselI(0,2*sqrt(n*(n - 1))*x).
%F A331657 a(n) = Sum_{k=0..n} binomial(n,k)^2 * n^k * (n - 1)^(n - k).
%F A331657 a(n) = P_n(2*n-1), where P_n is n-th Legendre polynomial.
%F A331657 a(n) = (-1)^n * 2F1(-n, n + 1; 1; n).
%F A331657 a(n) ~ 4^n * n^(n - 1/2) / (exp(1/2) * sqrt(Pi)). - _Vaclav Kotesovec_, Jan 26 2020
%F A331657 From _Seiichi Manyama_, Aug 30 2025: (Start)
%F A331657 a(n) = (-1)^n * Sum_{k=0..n} (1/(2*(2*n-1)))^(n-2*k) * binomial(-1/2,k) * binomial(k,n-k).
%F A331657 a(n) = Sum_{k=0..floor(n/2)} ((n-1)*n)^k * (2*n-1)^(n-2*k) * binomial(n,2*k) * binomial(2*k,k). (End)
%t A331657 Join[{1}, Table[Sum[(-1)^(n - k) Binomial[n, k] Binomial[n + k, k] n^k, {k, 0, n}], {n, 1, 16}]]
%t A331657 Table[SeriesCoefficient[1/Sqrt[1 - 2 (2 n - 1) x + x^2], {x, 0, n}], {n, 0, 16}]
%t A331657 Table[LegendreP[n, 2 n - 1], {n, 0, 16}]
%t A331657 Table[(-1)^n Hypergeometric2F1[-n, n + 1, 1, n], {n, 0, 16}]
%o A331657 (PARI) a(n) = {sum(k=0, n, (-1)^(n - k) * binomial(n,k) * binomial(n+k,k) * n^k)} \\ _Andrew Howroyd_, Jan 23 2020
%o A331657 (Magma) [&+[(-1)^(n-k)*Binomial(n,k)*Binomial(n+k,k)*n^k:k in [0..n]]:n in [0..16]]; // _Marius A. Burtea_, Jan 23 2020
%Y A331657 Cf. A001850, A006442, A084768, A084769, A110129, A331656, A335333.
%K A331657 nonn,changed
%O A331657 0,3
%A A331657 _Ilya Gutkovskiy_, Jan 23 2020