A294530 Binomial transform of A023871.

1, 2, 8, 33, 131, 497, 1834, 6635, 23622, 82942, 287656, 986552, 3349165, 11263951, 37558235, 124240204, 407951848, 1330340478, 4310385956, 13881618570, 44451643311, 141578435571, 448634389388, 1414774796929, 4441038400458, 13879652908322, 43197263002063

Offset: 0

Author

Vaclav Kotesovec, Nov 02 2017

Keywords

• nonn

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..2600

Crossrefs

Cf. A023871, A218481, A266232, A294500, A294529.

Programs

Mathematica

nmax = 40; s = CoefficientList[Series[Product[1/(1 - x^k)^(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) * A023871(k).

a(n) ~ exp(2^(5/4) * 3^(-5/4) * 5^(-1/4) * Pi * n^(3/4) + Pi^2 * sqrt(n) / (4*sqrt(30)) - Pi^3 * n^(1/4) / (32 * 2^(1/4) * 15^(3/4)) + Pi^4/3840 - Zeta(3)/(4*Pi^2)) * 2^(n - 7/8) / (15^(1/8) * n^(5/8)).

G.f.: (1/(1 - x))*exp(Sum_{k>=1} sigma_3(k)*x^k/(k*(1 - x)^k)). - Ilya Gutkovskiy, Aug 20 2018