OFFSET
0,11
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
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.
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]
CROSSREFS
KEYWORD
nonn
AUTHOR
Vaclav Kotesovec, Oct 15 2024
STATUS
approved