A255526 Coefficient of x^n in Product_{k>=1} 1/(1+x^k)^n.

-1, 1, -4, 17, -56, 172, -547, 1809, -6061, 20316, -68135, 229244, -774372, 2624119, -8912759, 30328593, -103382254, 352975681, -1206921212, 4132159452, -14163858895, 48601267199, -166930975524, 573872089212, -1974472043081, 6798561779868, -23425506369715

Offset: 1

Author

Vaclav Kotesovec, Feb 24 2015

Keywords

• sign

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 1..500

Crossrefs

Cf. A008485, A270913, A278428.

Programs

Mathematica

Table[SeriesCoefficient[Product[1/(1+x^k)^n,{k,1,n}],{x,0,n}],{n,1,30}]
(* Calculation of constant c: *) 1/Sqrt[(4 - r^2*s^3*Derivative[0, 2][QPochhammer][-1, r*s])*Pi] /. FindRoot[{QPochhammer[-1, r*s] == 2/s, 2/s + r*s*Derivative[0, 1][QPochhammer][-1, r*s] == 0}, {r, -1/3}, {s, 2}, WorkingPrecision -> 120] (* Vaclav Kotesovec, Oct 03 2023 *)

Formula

a(n) ~ (-1)^n * c * d^n / sqrt(n), where d = A318204 = 3.5097543279497033404372735..., c = 0.23322106096789389697797... .