A385092 G.f.: Sum_{k>=0} x^k * Product_{j=1..6*k} (1 + x^j)/(1 - x^j).

1, 1, 3, 7, 15, 29, 53, 93, 155, 251, 397, 613, 929, 1385, 2033, 2945, 4219, 5979, 8393, 11683, 16133, 22119, 30125, 40773, 54867, 73435, 97785, 129583, 170941, 224519, 293673, 382615, 496609, 642231, 827667, 1063073, 1361029, 1737081, 2210381, 2804485, 3548303, 4477229

Offset: 0

Author

Vaclav Kotesovec, Jun 17 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)/(1 - x^j), then a(n) ~ Gamma(1/m) * exp(Pi*sqrt(n)) / (m * 2^(2 + 2/m) * Pi^(1 - 1/m) * n^((m+1)/(2*m))).

Links

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

Crossrefs

Cf. A035298, A385070.

Cf. A207641, A385088, A385089, A385090, A385091.

Programs

Mathematica

nmax = 50; CoefficientList[Series[Sum[x^k*Product[(1+x^j)/(1-x^j), {j, 1, 6*k}], {k, 0, nmax}], {x, 0, nmax}], x]
nmax = 50; p = 1; q = 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]]]; q = Expand[q*(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))]; q = Take[q, Min[nmax + 1, Exponent[q, x] + 1, Length[q]]]; s += x^k*q/p;, {k, 1, nmax}]; CoefficientList[Series[s, {x, 0, nmax}], x]

Formula

a(n) ~ Gamma(1/6) * exp(Pi*sqrt(n)) / (3 * 2^(10/3) * Pi^(5/6) * n^(7/12)).