A010832 Expansion of Product_{k>=1} (1-x^k)^27.

1, -27, 324, -2223, 9126, -19278, -5967, 159030, -399087, 151593, 1270971, -2500875, 74970, 4203522, -1004157, -4796037, -11750778, 32885190, 10452375, -77533092, 27104868, 43070625, 63798840, -69960267, -215939061, 236414349, -37046646, 237487433, 85921371

Offset

0,2

References

Newman, Morris; A table of the coefficients of the powers of eta(tau). Nederl. Akad. Wetensch. Proc. Ser. A. 59 = Indag. Math. 18 (1956), 204-216.

Links

Seiichi Manyama, Table of n, a(n) for n = 0..10000

M. Boylan, Exceptional congruences for the coefficients of certain eta-product newforms, J. Number Theory 98 (2003), no. 2, 377-389.

Index entries for expansions of Product_{k >= 1} (1-x^k)^m

Formula

a(0) = 1, a(n) = -(27/n) * Sum_{k=1..n} A000203(k)*a(n-k) for n > 0. - Seiichi Manyama, Aug 13 2023

Empirical: Sum_{n>=0} a(n) / exp(n*Pi) = (1/2048) * exp(9*Pi/8) * Pi^(27/4) * 2^(7/8) / Gamma(3/4)^27 = A388227. - Simon Plouffe, Sep 15 2025

Mathematica

CoefficientList[Series[Product[(1-x^k)^27, {k, 30}], {x, 0, 30}], x] (* Harvey P. Dale, Feb 21 2026 *)

Crossrefs

Column k=27 of A286354.

Cf. A000203.

Sequence in context: A182130 A224311 A224375 * A022719 A290405 A048709

Adjacent sequences: A010829 A010830 A010831 * A010833 A010834 A010835

Keyword

sign,changed

Author

N. J. A. Sloane

Status

approved