OFFSET
0,31
COMMENTS
In general, if m > 0 and g.f. = Sum_{k>=1} x^(m*k^2) * Product_{j=1..k} (1 + x^j), then a(n) ~ (1+r) * exp(sqrt((4*m*(2*m+1)*log(r)^2 + 4*polylog(2, 1/(1+r)) - Pi^2/3)*n)) / (2*sqrt((r + 2*m*(1+r))*n)), where r is the smallest positive real root of the equation r^(2*m)*(1+r) = 1.
LINKS
Vaclav Kotesovec, Table of n, a(n) for n = 0..10000
Vaclav Kotesovec, Graph - the asymptotic ratio (100000 terms)
FORMULA
a(n) ~ (1+r) * exp(sqrt((84*log(r)^2 + 4*polylog(2, 1/(1+r)) - Pi^2/3)*n)) / (2*sqrt((6 + 7*r)*n)), where r = A230154 = 0.898653712628699293260875722... is the real root of the equation r^6*(1+r) = 1.
MATHEMATICA
nmax = 200; CoefficientList[Series[Sum[x^(3*k^2)*Product[1+x^j, {j, 1, k}], {k, 1, Sqrt[nmax/3]}], {x, 0, nmax}], x]
CROSSREFS
KEYWORD
nonn,look
AUTHOR
Vaclav Kotesovec, Oct 15 2024
STATUS
approved