A385070 G.f.: Sum_{k>=0} x^k * Product_{j=1..6*k} (1 + x^j).

1, 1, 2, 3, 5, 7, 10, 14, 18, 23, 30, 38, 48, 60, 74, 91, 112, 137, 166, 202, 244, 294, 352, 420, 500, 592, 700, 824, 968, 1133, 1323, 1541, 1791, 2077, 2403, 2776, 3198, 3679, 4226, 4845, 5546, 6340, 7236, 8246, 9385, 10667, 12108, 13728, 15545, 17581, 19860, 22409

Offset: 0

Author

Vaclav Kotesovec, Jun 16 2025

Keywords

• nonn

Comments

In general, for m>=1, if g.f. = Sum_{k>=0} x^k * Product_{j=1..m*k} (1 + x^j), then a(n) ~ Gamma(1/m) * 3^((m-2)/(4*m)) * exp(Pi*sqrt(n/3)) / (m * 2^(1 + 1/m) * Pi^(1 - 1/m) * n^((m+2)/(4*m))). - Vaclav Kotesovec, Jun 17 2025

Links

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

Crossrefs

Cf. A000009, A192433, A385067, A385068, A385069.

Cf. A035298.

Programs

Mathematica

nmax = 60; CoefficientList[Series[Sum[x^k*Product[1 + x^j, {j, 1, 6*k}], {k, 0, nmax}], {x, 0, nmax}], x]
nmax = 60; p = 1; s = 1; Do[p = Expand[p*(1 + x^(6*k))*(1 + x^(6*k - 1))*(1 + x^(6*k - 2))*(1 + x^(6*k - 3))*(1 + x^(6*k - 4))*(1 + x^(6*k - 5))]; 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(1/6) * exp(Pi*sqrt(n/3)) / (2^(13/6) * 3^(5/6) * Pi^(5/6) * n^(1/3)).