A258346 Expansion of Product_{k>=1} (1+x^k)^(k*(k-1)*(k-2)/6).

1, 0, 0, 1, 4, 10, 20, 39, 72, 144, 280, 567, 1112, 2187, 4204, 8073, 15309, 28986, 54548, 102286, 190881, 354717, 656194, 1208712, 2217624, 4052633, 7379630, 13390098, 24215587, 43649482, 78435884, 140513905, 250988186, 447037367, 794031641, 1406585604

Offset: 0

Author

Vaclav Kotesovec, May 27 2015

Keywords

• nonn

Links

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

Crossrefs

Cf. A248882, A028377, A258341, A258342, A258343, A258344, A258345, A258352.

Programs

Mathematica

nmax=50; CoefficientList[Series[Product[(1+x^k)^(k*(k-1)*(k-2)/6),{k,1,nmax}],{x,0,nmax}],x]

Formula

a(n) ~ (3*Zeta(5))^(1/10) / (2^(523/720) * 5^(2/5) * sqrt(Pi) * n^(3/5)) * exp(-2401 * Pi^16 / (10497600000000 * Zeta(5)^3) + 49*Pi^8 * Zeta(3) / (16200000 * Zeta(5)^2) - Zeta(3)^2 / (150*Zeta(5)) + (-343*Pi^12 / (2430000000 * 2^(3/5) * 15^(1/5) * Zeta(5)^(11/5)) + 7*Pi^4 * Zeta(3) / (4500 * 2^(3/5) * 15^(1/5) * Zeta(5)^(6/5))) * n^(1/5) + (-49*Pi^8 / (1080000 * 2^(1/5) * 15^(2/5) * Zeta(5)^(7/5)) + Zeta(3) / (2^(6/5) * (15*Zeta(5))^(2/5))) * n^(2/5) - 7*Pi^4 / (180 * 2^(4/5) * (15*Zeta(5))^(3/5)) * n^(3/5) + 5*(15*Zeta(5))^(1/5) / 2^(12/5) * n^(4/5)), where Zeta(3) = A002117, Zeta(5) = A013663.