A358362 a(n) = 16^n * Sum_{k=0..n} (-1)^k*binomial(-1/2, k)^2.
1, 12, 228, 3248, 56868, 846384, 14395920, 218556096, 3662534436, 56236646576, 933921124752, 14445103689408, 238434118702864, 3706773418885824, 60917716297733184, 950622015752780544, 15571249887287040804, 243694280206569964464, 3981466564018425521424
Offset: 0
Keywords
Programs
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Maple
a := n -> 16^n*add((-1)^k*binomial(-1/2, k)^2, k = 0..n): seq(a(n), n = 0..19);
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Mathematica
a[n_] := 16^n * Sum[(-1)^k*Binomial[-1/2, k]^2, {k, 0, n}]; Array[a, 19, 0] (* Amiram Eldar, Nov 12 2022 *)
Formula
a(n) = (64*(2*n - 1)^2*a(n - 2) + (16*n - 4)*a(n - 1)) / n^2.
G.f.: hypergeom([1/2, 1/2], [1], -16*x)/(16*x - 1).
G.f.: 2*EllipticK(4*I*sqrt(x))/(Pi*(1 - 16*x)).
a(n) ~ A014549 * 2^(4*n). - Vaclav Kotesovec, Nov 14 2023