A294529 Binomial transform of A001156.

1, 2, 4, 8, 17, 38, 86, 192, 420, 905, 1939, 4163, 8987, 19494, 42368, 91990, 199127, 429345, 921982, 1972553, 4206909, 8949412, 19001874, 40293048, 85373962, 180826115, 382957231, 811027414, 1717497958, 3636335170, 7695599294, 16275268520, 34389570596

Offset: 0

Author

Vaclav Kotesovec, Nov 02 2017

Keywords

• nonn

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..3000

Crossrefs

Cf. A001156, A218481, A266232, A294500, A294530.

Programs

Mathematica

nmax = 40; s = CoefficientList[Series[Product[1/(1 - x^(k^2)), {k, 1, nmax}], {x, 0, nmax}], x]; Table[Sum[Binomial[n, k] * s[[k+1]], {k, 0, n}], {n, 0, nmax}]

Formula

a(n) = Sum_{k=0..n} binomial(n,k) * A001156(k).

a(n) ~ exp(3 * 2^(-5/3) * Pi^(1/3) * Zeta(3/2)^(2/3) * n^(1/3)) * Zeta(3/2)^(2/3) * 2^(n - 7/6) / (sqrt(3) * Pi^(7/6) * n^(7/6)).

G.f.: (1/(1 - x))*Product_{k>=1} 1/(1 - x^(k^2)/(1 - x)^(k^2)). - Ilya Gutkovskiy, Aug 20 2018