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A258351
Expansion of Product_{k>=1} 1/(1-x^k)^(k*(k-1)*(k-2)).
8
1, 0, 0, 6, 24, 60, 141, 354, 996, 2720, 7194, 18306, 46154, 115506, 288195, 713210, 1749732, 4253148, 10259302, 24573390, 58491312, 138371354, 325415727, 760899396, 1769420183, 4093054602, 9420739965, 21578842582, 49199229066, 111672215658, 252381169048
OFFSET
0,4
LINKS
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]
KEYWORD
nonn
AUTHOR
Vaclav Kotesovec, May 27 2015
STATUS
approved