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

1,3

Links

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

Formula

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

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 *)

Crossrefs

Cf. A008485, A270913, A278428.

Sequence in context: A157492 A108140 A213577 * A264218 A121327 A317757

Adjacent sequences: A255523 A255524 A255525 * A255527 A255528 A255529

Keyword

sign

Author

Vaclav Kotesovec, Feb 24 2015

Status

approved