A000728 - OEIS

A000728

Expansion of Product_{n>=1} (1-x^n)^5.
(Formerly M3742 N1529)

1, -5, 5, 10, -15, -6, -5, 25, 15, -20, 9, -45, -5, 25, 20, 10, 15, 20, -50, -35, -30, 55, -50, 15, 80, 1, 50, -35, -45, -15, 5, -50, -25, -55, 85, 51, 50, 10, -40, 65, 10, -10, -115, 50, -115, -100, 85, 80, -30, 5, 20, 45, 70, 65, 45, -55, -100

(list; graph; refs; listen; history; text; internal format)

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) = -(5/n)*Sum_{k=1..n} A000203(k)*a(n-k) for n > 0. - Seiichi Manyama, Mar 26 2017

G.f.: exp(-5*Sum_{k>=1} x^k/(k*(1 - x^k))). - Ilya Gutkovskiy, Feb 05 2018

Empirical: Sum_{n>=0} a(n) / exp(n*Pi) = (1/4) * exp(5 * Pi / 24) * Pi^(5/4) * 2^(1/8) / Gamma(3/4)^5 = A388088. - Simon Plouffe, Sep 14 2025

MATHEMATICA

CoefficientList[QPochhammer[x]^5 + O[x]^60, x] (* Jean-François Alcover, Feb 10 2016 *)

CROSSREFS

Cf. A258405.

Sequence in context: A285932 A109064 A138506 * A242895 A242129 A022088

Adjacent sequences: A000725 A000726 A000727 * A000729 A000730 A000731

KEYWORD

sign

AUTHOR

N. J. A. Sloane

STATUS

approved