A258350 Expansion of Product_{k>=1} 1/(1-x^k)^(k*(k+1)*(k+2)).

1, 6, 45, 260, 1410, 7026, 33212, 149190, 643959, 2681020, 10820736, 42468828, 162566956, 608302638, 2229485529, 8016901068, 28324233846, 98447346282, 336996263702, 1137220855428, 3786525025002, 12449461237388, 40446207528429, 129926295916884, 412912082761651

Offset

0,2

Links

Vaclav Kotesovec, Table of n, a(n) for n = 0..1000

Formula

a(n) ~ (3*Zeta(5))^(79/600) / (2^(21/200) * sqrt(5*Pi) * n^(379/600)) * exp(2*Zeta'(-1) - 3*Zeta(3)/(4*Pi^2) - Pi^16 / (518400000 * Zeta(5)^3) + Pi^8 * Zeta(3) / (36000 * Zeta(5)^2) - Zeta(3)^2 / (15*Zeta(5)) + Zeta'(-3) + (Pi^12 / (1800000 * 2^(3/5) * 3^(1/5) * Zeta(5)^(11/5)) - Pi^4 * Zeta(3) / (150 * 2^(3/5) * 3^(1/5) * Zeta(5)^(6/5))) * n^(1/5) + (-Pi^8 / (12000 * 2^(1/5) * 3^(2/5) * Zeta(5)^(7/5)) + Zeta(3) / (2^(1/5) * (3*Zeta(5))^(2/5))) * n^(2/5) + Pi^4 / (30 * 2^(4/5) * (3*Zeta(5))^(3/5)) * n^(3/5) + 5 * (3*Zeta(5))^(1/5) / 2^(7/5) * n^(4/5)), where Zeta(3) = A002117, Zeta(5) = A013663, Zeta'(-1) = A084448 = 1/12 - log(A074962), Zeta'(-3) = ((gamma + log(2*Pi) - 11/6)/30 - 3*Zeta'(4)/Pi^4)/4.

Mathematica

nmax=40; CoefficientList[Series[Product[1/(1-x^k)^(k*(k+1)*(k+2)), {k, 1, nmax}], {x, 0, nmax}], x]

Crossrefs

Cf. A023872, A000335, A258342, A258347, A258348, A258349, A258351, A258352.

Sequence in context: A054563 A288835 A162230 * A258623 A003486 A262415

Adjacent sequences: A258347 A258348 A258349 * A258351 A258352 A258353

Keyword

nonn

Author

Vaclav Kotesovec, May 27 2015

Status

approved