A000730 Expansion of Product_{n>=1} (1 - x^n)^7. (Formerly M4347 N1821)

1, -7, 14, 7, -49, 21, 35, 41, -49, -133, 98, -21, 126, 112, -176, -105, -126, 140, -35, 147, 259, 98, -420, -224, 238, -455, 273, -14, 322, 406, -35, -7, -637, -196, 245, -181, -574, 462, 147, 924, 217, -329, -140, -7, -371, -777

Offset

0,2

References

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

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

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

Matthew Boylan, Exceptional congruences for the coefficients of certain eta-product newforms, J. Number Theory 98 (2003), no. 2, 377-389. MR1955423 (2003k:11071)

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

Simon Plouffe, Numbers in the base e^Pi, arXiv:2509.15609 [math.NT], 2025. See p. 12/24, marked 9.

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

Formula

a(0) = 1, a(n) = -(7/n)*Sum_{k=1..n} A000203(k)*a(n-k) for n > 0. - Seiichi Manyama, Mar 26 2017

G.f.: exp(-7*Sum_{k>=1} x^k/(k*(1 - x^k))). - Ilya Gutkovskiy, Feb 05 2018

Empirical: Sum_{n>=0} a(n) / exp(n*Pi) = (1/8) * exp(7*Pi/24) * Pi^(7/4) * 2^(3/8) / Gamma(3/4)^7 = A388090. - Simon Plouffe, Sep 14 2025

Mathematica

CoefficientList[QPochhammer[x]^7 + O[x]^50, x] (* Jean-François Alcover, Feb 10 2016 *)

Crossrefs

Sequence in context: A115146 A340312 A029844 * A160534 A022699 A362586

Adjacent sequences: A000727 A000728 A000729 * A000731 A000732 A000733

Keyword

sign

Author

N. J. A. Sloane

Status

approved