A358363 a(n) = 16^n * Sum_{k=0..n} (-1)^k*binomial(1/2, k)^2.

1, 12, 196, 3120, 50020, 799536, 12799632, 204724416, 3276326820, 52413049520, 838703348496, 13418125153472, 214703825630736, 3435088134123200, 54963617747611200, 879389273444524800, 14070604335190692900, 225124668703739770800, 3602061930346132909200

Offset: 0

Author

Peter Luschny, Nov 12 2022

Keywords

• nonn

Crossrefs

Cf. A358362, A358364, A358365, A367331.

Programs

Maple

a := n -> 16^n*add((-1)^k*binomial(1/2, k)^2, k = 0..n):
seq(a(n), n = 0..18);

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 - 3)^2*a(n - 2) + 12*(4*n - 3)*a(n - 1)) / n^2.

G.f.: hypergeom([-1/2, -1/2], [1], -16*x)/(1 - 16*x).

a(n) ~ sqrt(Pi) * 2^(4*n + 5/2) / Gamma(1/4)^2. - Vaclav Kotesovec, Nov 14 2023