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%I #10 Feb 17 2024 14:55:16
%S 1,56,3288,197312,11992024,734961216,45312662976,2806150276608,
%T 174385474327512,10867238335817024,678767129043750208,
%U 42476876703235742208,2662498434919062169024,167121637293079702800896,10502764033533202152955392,660751064709823030602903552
%N a(n) = 64^n * hypergeometric([1/2, 1/2, 1/2, -n], [1, 1, 1], 1).
%F From _Vaclav Kotesovec_, Feb 17 2024: (Start)
%F Recurrence: n^3*a(n) = 8*(2*n - 1)*(12*n^2 - 12*n + 7)*a(n-1) - 3072*(n-1)*(4*n^2 - 8*n + 5)*a(n-2) + 131072*(n-2)*(n-1)*(2*n - 3)*a(n-3).
%F a(n) ~ 2^(6*n-1) * log(n)^2 / (Pi^(5/2)*sqrt(n)) * (1 + c1/log(n) + c2/log(n)^2), where c1 = 12.24478621876219067188873812349562995129232082... and c2 = 32.54889518525243748904367845713571175154193233... (End)
%p a := n -> 64^n*hypergeom([1/2, 1/2, 1/2, -n], [1, 1, 1], 1):
%p seq(simplify(a(n)), n = 0..15);
%t a[n_] := 64^n * HypergeometricPFQ[{1/2, 1/2, 1/2, -n}, {1, 1, 1}, 1]; Array[a, 16, 0] (* _Amiram Eldar_, Nov 12 2022 *)
%Y Cf. A358116, A358117.
%K nonn
%O 0,2
%A _Peter Luschny_, Nov 12 2022
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