A264905 Expansion of Product_{k>=1} (1 + x^k + x^(3*k)).

1, 1, 1, 3, 2, 4, 6, 7, 8, 13, 16, 18, 26, 29, 38, 49, 58, 68, 90, 101, 125, 156, 181, 214, 263, 304, 358, 435, 505, 589, 701, 812, 939, 1115, 1275, 1485, 1736, 1991, 2286, 2667, 3038, 3489, 4028, 4588, 5240, 6036, 6833, 7787, 8904, 10078, 11429, 13020, 14698

Offset: 0

Author

Vaclav Kotesovec, Nov 28 2015

Keywords

• nonn

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..5000

Crossrefs

Cf. A000009, A000726, A001935, A100405, A266647, A266648, A266649, A266650, A266686, A275820.

Programs

Mathematica

nmax = 100; CoefficientList[Series[Product[1+x^k+x^(3*k), {k,1,nmax}], {x,0,nmax}], x]
nmax = 100; p = ConstantArray[0, nmax + 1]; p[[1]] = 1; p[[2]] = 1; p[[4]] = 1; Do[Do[p[[j+1]] = p[[j+1]] + p[[j - k + 1]] + If[j < 3*k, 0, p[[j - 3*k + 1]]], {j, nmax, k, -1}];, {k, 2, nmax}]; p (* Vaclav Kotesovec, May 10 2018 *)

Formula

a(n) ~ c^(1/4) * exp(2*sqrt(c*n)) / (2*sqrt(3*Pi)*n^(3/4)), where c = Integral_{0..infinity} log(1 + exp(-x) + exp(-3*x)) dx = 0.9953865985263189816963357718655148864441174218433250148867... . - Vaclav Kotesovec, Jan 05 2016