|
|
A261389
|
|
Expansion of Product_{k>=1} ((1+x^k)/(1-x^k))^(3*k).
|
|
6
|
|
|
1, 6, 30, 128, 486, 1704, 5604, 17484, 52206, 150118, 417696, 1128984, 2973476, 7650720, 19272432, 47616568, 115570014, 275921460, 648771802, 1503889488, 3439990344, 7770915816, 17349229908, 38306180052, 83694778556, 181052778078, 387976101432, 823939048560
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
0,2
|
|
COMMENTS
|
In general, if g.f. = Product_{k>=1} ((1+x^k)/(1-x^k))^(t*k) and t>=1, then a(n) ~ exp(t/12 + 3/2 * (7*t*Zeta(3)/2)^(1/3) * n^(2/3)) * t^(1/6 + t/36) * (7*Zeta(3))^(1/6 + t/36) / (A^t * 2^(2/3 + t/9) * sqrt(3*Pi) * n^(2/3 + t/36)), where Zeta(3) = A002117 and A = A074962 is the Glaisher-Kinkelin constant.
|
|
LINKS
|
|
|
FORMULA
|
a(n) ~ exp(1/4 + 3/2 * (21*Zeta(3)/2)^(1/3) * n^(2/3)) * (7*Zeta(3)/3)^(1/4) / (2 * A^3 * sqrt(Pi) * n^(3/4)), where Zeta(3) = A002117 and A = A074962 is the Glaisher-Kinkelin constant.
|
|
MATHEMATICA
|
nmax = 40; CoefficientList[Series[Product[((1+x^k)/(1-x^k))^(3*k), {k, 1, nmax}], {x, 0, nmax}], x]
|
|
CROSSREFS
|
|
|
KEYWORD
|
nonn
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|