A385069 G.f.: Sum_{k>=0} x^k * Product_{j=1..5*k} (1 + x^j).

1, 1, 2, 3, 5, 7, 10, 13, 17, 22, 28, 36, 45, 56, 69, 86, 106, 130, 159, 192, 232, 279, 333, 397, 472, 557, 657, 773, 905, 1059, 1236, 1437, 1669, 1935, 2236, 2582, 2976, 3422, 3930, 4507, 5157, 5894, 6728, 7664, 8721, 9911, 11246, 12746, 14430, 16312, 18421, 20780

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, A385067, A385068, A385070.

Cf. A035297.

Programs

Mathematica

nmax = 60; CoefficientList[Series[Sum[x^k*Product[1 + x^j, {j, 1, 5*k}], {k, 0, nmax}], {x, 0, nmax}], x]
nmax = 60; p = 1; s = 1; Do[p = Expand[p*(1 + x^(5*k))*(1 + x^(5*k - 1))*(1 + x^(5*k - 2))*(1 + x^(5*k - 3))*(1 + x^(5*k - 4))]; 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(1/5) * 3^(3/20) * exp(Pi*sqrt(n/3)) / (5 * 2^(6/5) * Pi^(4/5) * n^(7/20)).