A385067 G.f.: Sum_{k>=0} x^k * Product_{j=1..3*k} (1 + x^j).

1, 1, 2, 3, 5, 6, 8, 11, 14, 18, 23, 30, 38, 47, 58, 71, 87, 106, 128, 154, 185, 221, 263, 313, 370, 437, 514, 603, 705, 822, 958, 1112, 1289, 1491, 1721, 1982, 2279, 2617, 2999, 3432, 3921, 4473, 5095, 5795, 6583, 7468, 8461, 9574, 10820, 12214, 13772, 15512, 17453

Offset: 0

Author

Vaclav Kotesovec, Jun 16 2025

Keywords

• nonn

Links

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

Crossrefs

Cf. A000009, A192433, A385068, A385069, A385070.

Cf. A035295.

Programs

Mathematica

nmax = 60; CoefficientList[Series[Sum[x^k*Product[1 + x^j, {j, 1, 3*k}], {k, 0, nmax}], {x, 0, nmax}], x]
nmax = 60; p = 1; s = 1; Do[p = Expand[p*(1 + x^(3*k))*(1 + x^(3*k - 1))*(1 + x^(3*k - 2))]; p = Take[p, Min[nmax + 1, Exponent[p, x] + 1, Length[p]]]; s += p*x^k;, {k, 1, nmax}]; CoefficientList[Series[s, {x, 0, nmax}], x]

Formula

a(n) ~ Gamma(1/3) * exp(Pi*sqrt(n/3)) / (2^(4/3) * 3^(11/12) * Pi^(2/3) * n^(5/12)).