A266647 Expansion of Product_{k>=1} (1 + x^k + x^(3*k)) / (1 - x^k).

1, 2, 4, 9, 15, 27, 46, 75, 118, 187, 285, 429, 639, 935, 1354, 1945, 2758, 3878, 5417, 7493, 10300, 14070, 19087, 25741, 34542, 46081, 61185, 80869, 106391, 139368, 181867, 236357, 306060, 394939, 507860, 650946, 831792, 1059600, 1345920, 1704880, 2153682

Offset: 0

Author

Vaclav Kotesovec, Jan 02 2016

Keywords

• nonn

Comments

Convolution of A264905 and A000041.

Links

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

Crossrefs

Cf. A100405, A264905, A266648, A266649, A266650.

Programs

Mathematica

nmax = 50; CoefficientList[Series[Product[(1+x^k+x^(3*k))/(1-x^k), {k, 1, nmax}], {x, 0, nmax}], x]

Formula

a(n) ~ sqrt(6*c + Pi^2) * exp(sqrt((4*c + 2*Pi^2/3)*n)) / (12*Pi*n), where c = Integral_{0..infinity} log(1 + exp(-x) + exp(-3*x)) dx = 0.9953865985263189816963357718655148864441174218433250148867... . - Vaclav Kotesovec, Jan 05 2016