A258345 Expansion of Product_{k>=1} (1+x^k)^(k*(k-1)*(k-2)).

1, 0, 0, 6, 24, 60, 135, 354, 972, 2684, 6990, 17802, 44627, 111582, 277329, 684164, 1671984, 4050096, 9735209, 23238480, 55120950, 129940442, 304502583, 709464798, 1643920584, 3789158988, 8690016942, 19833550266, 45056952957, 101900481462, 229462378987

Offset

0,4

Links

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

Formula

a(n) ~ 3^(1/5) * Zeta(5)^(1/10) / (2^(91/120) * 5^(2/5) * sqrt(Pi) * n^(3/5)) * exp(-2401 * Pi^16 / (1749600000000*Zeta(5)^3) + 49 * Pi^8 * Zeta(3) / (2700000 * Zeta(5)^2) - Zeta(3)^2 / (25*Zeta(5)) + (-343 * Pi^12 / (405000000 * 2^(4/5) * 3^(2/5) * 5^(1/5) * Zeta(5)^(11/5)) + 7*Pi^4 * Zeta(3) / (750 * 2^(4/5) * 3^(2/5) * 5^(1/5) * Zeta(5)^(6/5))) * n^(1/5) + (-49*Pi^8 / (180000 * 2^(3/5) * 3^(4/5) * 5^(2/5) * Zeta(5)^(7/5)) + 3^(1/5) * Zeta(3) / (2^(3/5) * (5*Zeta(5))^(2/5))) * n^(2/5) - 7*Pi^4 / (180 * 2^(2/5) * 3^(1/5) * (5*Zeta(5))^(3/5)) * n^(3/5) + 5*3^(2/5) * ((5*Zeta(5))/2)^(1/5)/4 * n^(4/5)), where Zeta(3) = A002117, Zeta(5) = A013663.

Mathematica

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

Crossrefs

Cf. A248882, A028377, A258341, A258342, A258343, A258344, A258346, A258351.

Sequence in context: A331433 A329119 A371044 * A258351 A130669 A214308

Adjacent sequences: A258342 A258343 A258344 * A258346 A258347 A258348

Keyword

nonn

Author

Vaclav Kotesovec, May 27 2015

Status

approved