A358363 - OEIS

%I #20 Nov 15 2023 03:12:34

%S 1,12,196,3120,50020,799536,12799632,204724416,3276326820,52413049520,

%T 838703348496,13418125153472,214703825630736,3435088134123200,

%U 54963617747611200,879389273444524800,14070604335190692900,225124668703739770800,3602061930346132909200

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

%F a(n) = (64*(2*n - 3)^2*a(n - 2) + 12*(4*n - 3)*a(n - 1)) / n^2.

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

%F a(n) ~ sqrt(Pi) * 2^(4*n + 5/2) / Gamma(1/4)^2. - _Vaclav Kotesovec_, Nov 14 2023

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

%p seq(a(n), n = 0..18);

%t a[n_] := 16^n * Sum[(-1)^k*Binomial[1/2, k]^2, {k, 0, n}]; Array[a, 19, 0] (* _Amiram Eldar_, Nov 12 2022 *)

%Y Cf. A358362, A358364, A358365, A367331.

%K nonn

%O 0,2

%A _Peter Luschny_, Nov 12 2022