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A331656 a(n) = Sum_{k=0..n} binomial(n,k) * binomial(n+k,k) * n^k.

%I A331656 #21 Aug 30 2025 10:13:09
%S A331656 1,3,37,847,28401,1256651,69125869,4548342975,348434664769,
%T A331656 30463322582899,2993348092318101,326572612514776079,
%U A331656 39170287549040392369,5123157953193993402171,725662909285939100555101,110662236267661479984580351,18077209893508013563092846849
%N A331656 a(n) = Sum_{k=0..n} binomial(n,k) * binomial(n+k,k) * n^k.
%H A331656 Seiichi Manyama, <a href="/A331656/b331656.txt">Table of n, a(n) for n = 0..321</a>
%H A331656 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/LegendrePolynomial.html">Legendre Polynomial</a>
%F A331656 a(n) = central coefficient of (1 + (2*n + 1)*x + n*(n + 1)*x^2)^n.
%F A331656 a(n) = [x^n] 1 / sqrt(1 - 2*(2*n + 1)*x + x^2).
%F A331656 a(n) = n! * [x^n] exp((2*n + 1)*x) * BesselI(0,2*sqrt(n*(n + 1))*x).
%F A331656 a(n) = Sum_{k=0..n} binomial(n,k)^2 * n^k * (n + 1)^(n - k).
%F A331656 a(n) = P_n(2*n+1), where P_n is n-th Legendre polynomial.
%F A331656 a(n) ~ exp(1/2) * 4^n * n^(n - 1/2) / sqrt(Pi). - _Vaclav Kotesovec_, Jan 28 2020
%F A331656 From _Seiichi Manyama_, Aug 30 2025: (Start)
%F A331656 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 A331656 a(n) = Sum_{k=0..floor(n/2)} (n*(n+1))^k * (2*n+1)^(n-2*k) * binomial(n,2*k) * binomial(2*k,k). (End)
%t A331656 Join[{1}, Table[Sum[Binomial[n, k] Binomial[n + k, k] n^k, {k, 0, n}], {n, 1, 16}]]
%t A331656 Table[SeriesCoefficient[1/Sqrt[1 - 2 (2 n + 1) x + x^2], {x, 0, n}], {n, 0, 16}]
%t A331656 Table[LegendreP[n, 2 n + 1], {n, 0, 16}]
%t A331656 Table[Hypergeometric2F1[-n, n + 1, 1, -n], {n, 0, 16}]
%o A331656 (PARI) a(n) = {sum(k=0, n, binomial(n,k) * binomial(n+k,k) * n^k)} \\ _Andrew Howroyd_, Jan 23 2020
%Y A331656 Main diagonal of A335333.
%Y A331656 Cf. A001850, A006442, A084768, A084769, A110129, A331657.
%K A331656 nonn,changed
%O A331656 0,2
%A A331656 _Ilya Gutkovskiy_, Jan 23 2020