A377075 G.f.: Sum_{k>=0} x^(8*k^2) / Product_{j=1..8*k-1} (1 - x^j).

0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 2, 3, 5, 7, 11, 15, 21, 28, 38, 49, 65, 82, 105, 131, 164, 201, 248, 300, 364, 436, 522, 618, 734, 861, 1011, 1178, 1372, 1586, 1835, 2108, 2422, 2768, 3162, 3595, 4088, 4627, 5237, 5907, 6660, 7485, 8414, 9429, 10568, 11817, 13213

Offset: 0

Author

Vaclav Kotesovec, Oct 15 2024

Keywords

• nonn

Comments

In general, for m >= 1, if g.f.= Sum_{k>=1} x^(m*k^2)/Product_{j=1..m*k-1} (1-x^j), then a(n) ~ r^2 * (m*log(r)^2 + polylog(2, r^2))^(1/4) * exp(2*sqrt((m*log(r)^2 + polylog(2, r^2))*n)) / (2*sqrt(Pi*m*(m - (m-2)*r^2)) * n^(3/4)), where r is the positive real root of the equation r^2 = 1 - r^m.

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..10000

Crossrefs

Column 8 of A350889.

Cf. A376658.

Programs

Mathematica

nmax = 100; CoefficientList[Series[Sum[x^(8*k^2)/Product[1-x^j, {j, 1, 8*k-1}], {k, 1, Sqrt[nmax/8]}], {x, 0, nmax}], x]

Formula

Limit_{n->oo} a(n)^(1/sqrt(n)) = A376658.

a(n) ~ r^2 * (8*log(r)^2 + polylog(2, r^2))^(1/4) * exp(2*sqrt((8*log(r)^2 + polylog(2, r^2))*n)) / (8*sqrt(Pi*(4 - 3*r^2)) * n^(3/4)), where r = 0.8511709340670154789... is the positive real root of the equation r^2 = 1 - r^8.