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%I #17 Nov 15 2023 03:11:19
%S 1,20,356,6096,102436,1702480,28093456,461273920,7546019620,
%T 123100218320,2003738272656,32557446669120,528231606378256,
%U 8559878182412096,138567392514153536,2241139725237406976,36219533239041063716,584958249814679707856,9441690077748181415696
%N a(n) = 16^n * Sum_{k=0..n} binomial(-1/2, k)^2.
%F a(n) = (-64*(2*n - 1)^2*a(n - 2) + 4*(8*n^2 - 4*n + 1)*a(n - 1)) / n^2.
%F G.f.: hypergeom([1/2, 1/2], [1], 16*x)/(1 - 16*x).
%F G.f.: 2*EllipticK(4*sqrt(x))/(Pi*(1 - 16*x)).
%F a(n) ~ (log(n) + gamma + 4*log(2)) * 2^(4*n)/Pi, where gamma is the Euler-Mascheroni constant A001620. - _Vaclav Kotesovec_, Nov 14 2023
%p a := n -> 16^n*add(binomial(-1/2, k)^2, k = 0..n):
%p seq(a(n), n = 0..18);
%t a[n_] := 16^n * Sum[Binomial[-1/2, k]^2, {k, 0, n}]; Array[a, 19, 0] (* _Amiram Eldar_, Nov 12 2022 *)
%Y Cf. A358362, A358363, A358364, A367333.
%K nonn
%O 0,2
%A _Peter Luschny_, Nov 12 2022