A358364 a(n) = 16^n * Sum_{k=0..n} binomial(1/2, k)^2.
1, 20, 324, 5200, 83300, 1333584, 21344400, 341580096, 5466017700, 87464462800, 1399525960976, 22393543798080, 358310523944464, 5733141459080000, 91732470946920000, 1467748145667974400, 23484346290765886500, 375754541311565499600, 6012139892071344570000
Offset: 0
Keywords
Programs
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Maple
a := n -> 16^n*add(binomial(1/2, k)^2, k = 0..n): seq(a(n), n = 0..18);
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Mathematica
a[n_] := 16^n * Sum[Binomial[1/2, k]^2, {k, 0, n}]; Array[a, 19, 0] (* Amiram Eldar, Nov 12 2022 *)
Formula
a(n) = (16*n + 4)*(2*n - 1)^2*a(n - 1) / ((4*n - 3) * n^2).
G.f.: hypergeom([-1/2, -1/2], [1], 16*x)/(1 - 16*x).
a(n) ~ 2^(4*n+2) / Pi. - Vaclav Kotesovec, Nov 14 2023