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a(n) = Sum_{k=0..n} binomial(n,k) * binomial(n+k,k) * n^k.
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%I #13 Jun 02 2020 01:15:57

%S 1,3,37,847,28401,1256651,69125869,4548342975,348434664769,

%T 30463322582899,2993348092318101,326572612514776079,

%U 39170287549040392369,5123157953193993402171,725662909285939100555101,110662236267661479984580351,18077209893508013563092846849

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

%H Seiichi Manyama, <a href="/A331656/b331656.txt">Table of n, a(n) for n = 0..321</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/LegendrePolynomial.html">Legendre Polynomial</a>

%F a(n) = central coefficient of (1 + (2*n + 1)*x + n*(n + 1)*x^2)^n.

%F a(n) = [x^n] 1 / sqrt(1 - 2*(2*n + 1)*x + x^2).

%F a(n) = n! * [x^n] exp((2*n + 1)*x) * BesselI(0,2*sqrt(n*(n + 1))*x).

%F a(n) = Sum_{k=0..n} binomial(n,k)^2 * n^k * (n + 1)^(n - k).

%F a(n) = P_n(2*n+1), where P_n is n-th Legendre polynomial.

%F a(n) ~ exp(1/2) * 4^n * n^(n - 1/2) / sqrt(Pi). - _Vaclav Kotesovec_, Jan 28 2020

%t Join[{1}, Table[Sum[Binomial[n, k] Binomial[n + k, k] n^k, {k, 0, n}], {n, 1, 16}]]

%t Table[SeriesCoefficient[1/Sqrt[1 - 2 (2 n + 1) x + x^2], {x, 0, n}], {n, 0, 16}]

%t Table[LegendreP[n, 2 n + 1], {n, 0, 16}]

%t Table[Hypergeometric2F1[-n, n + 1, 1, -n], {n, 0, 16}]

%o (PARI) a(n) = {sum(k=0, n, binomial(n,k) * binomial(n+k,k) * n^k)} \\ _Andrew Howroyd_, Jan 23 2020

%Y Cf. A001850, A006442, A084768, A084769, A110129, A331657.

%K nonn

%O 0,2

%A _Ilya Gutkovskiy_, Jan 23 2020