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

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

Author

N. J. A. Sloane

Keywords

• sign

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.
• 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
• M. Boylan, Exceptional congruences for the coefficients of certain eta-product newforms, J. Number Theory 98 (2003), no. 2, 377-389. MR1955423 (2003k:11071)
• M. 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]
• Index entries for expansions of Product_{k >= 1} (1-x^k)^m

Programs

Mathematica

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

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