OFFSET
0,3
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) * exp(Pi*sqrt(2*n/3)) / (m * 2^((3*m + 1)/(2*m)) * 3^(1/(2*m)) * Pi^(1 - 1/m) * n^((m+1)/(2*m))). - Vaclav Kotesovec, Jun 17 2025
LINKS
Vaclav Kotesovec, Table of n, a(n) for n = 0..5000
FORMULA
a(n) ~ Gamma(1/6) * exp(Pi*sqrt(2*n/3)) / (2^(31/12) * 3^(13/12) * Pi^(5/6) * n^(7/12)). - Vaclav Kotesovec, Jun 17 2025
MATHEMATICA
nmax = 50; CoefficientList[Series[Sum[x^k/Product[1 - x^j, {j, 1, 6*k}], {k, 0, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Jun 16 2025 *)
nmax = 50; 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+=x^k/p; , {k, 1, nmax}]; CoefficientList[Series[s, {x, 0, nmax}], x] (* Vaclav Kotesovec, Jun 16 2025 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
EXTENSIONS
More terms from Vaclav Kotesovec, Jun 16 2025
STATUS
approved
