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%I #13 Mar 18 2024 06:06:30
%S 1,12,132,1200,5220,-132048,-5451376,-139104576,-3034129500,
%T -61171843920,-1176294856176,-21916435874112,-399241706218992,
%U -7151078337480000,-126420386691188160,-2211675290036790528,-38363623542890191836,-660751288131343246416,-11312478475520480652400
%N a(n) = 16^n * Sum_{k=0..n} (-1)^k * binomial(1/2, k)^2 * binomial(n, k).
%C Belongs to the family of Apéry-like sequences.
%F a(n) = 16^n * hypergeom([-1/2, -1/2, -n], [1, 1], 1).
%F From _Vaclav Kotesovec_, Nov 12 2022: (Start)
%F G.f.: LegendreP(1/2, 1 - 32*x) / (1 - 16*x)^(3/2).
%F Recurrence: n^2*a(n) = 4*(8*n^2 - 5)*a(n-1) - 256*(n-1)*(n+1)*a(n-2).
%F a(n) ~ -2^(4*n + 1) * sqrt(n) * log(n) / Pi^(3/2) * (1 - c/log(n)), where c = 1.2639012517387952828900951811685381605048398578985... (End)
%F c = 6 - 6*log(2) - gamma, where gamma is the Euler-Mascheroni constant A001620. - _Vaclav Kotesovec_, Mar 18 2024
%p a := n -> 16^n*add((-1)^k*binomial(1/2, k)^2*binomial(n, k), k = 0..n):
%p seq(a(n), n = 0..18);
%t a[n_] := 16^n * HypergeometricPFQ[{-1/2, -1/2, -n}, {1, 1}, 1]; Array[a, 19, 0] (* _Amiram Eldar_, Nov 12 2022 *)
%t CoefficientList[Series[LegendreP[1/2, 1 - 32*x]/(1 - 16*x)^(3/2), {x, 0, 20}], x] (* _Vaclav Kotesovec_, Nov 12 2022 *)
%Y Cf. A358108, A358109, A143583.
%K sign
%O 0,2
%A _Peter Luschny_, Nov 12 2022
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