A266686 Expansion of Product_{k>=1} (1 + x^k - x^(3*k)).

1, 1, 1, 1, 2, 2, 2, 3, 4, 5, 4, 6, 8, 9, 10, 11, 14, 16, 18, 21, 25, 28, 31, 36, 41, 48, 52, 59, 69, 77, 85, 96, 109, 121, 133, 151, 172, 189, 208, 231, 260, 287, 316, 350, 390, 432, 471, 521, 578, 636, 695, 764, 842, 924, 1009, 1107, 1218, 1330, 1449, 1584

Offset: 0

Author

Vaclav Kotesovec, Jan 02 2016

Keywords

• nonn

Links

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

Crossrefs

Cf. A264905, A266647, A266650, A275820.

Programs

Mathematica

nmax=60; 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(Pi)*n^(3/4)), where c = Integral_{0..infinity} log(1 + exp(-x) - exp(-3*x)) dx = 0.59698046904738615106237970379036510874974380079287087827737... . - Vaclav Kotesovec, Jan 05 2016