A030181 Expansion of (eta(q) / eta(q^7))^4 in powers of q.

1, -4, 2, 8, -5, -4, -10, 12, -7, 8, 46, -36, -26, -44, 46, -28, 42, 188, -132, -96, -167, 172, -98, 120, 596, -420, -286, -492, 496, -280, 368, 1680, -1151, -792, -1332, 1320, -735, 916, 4264, -2908, -1960, -3252, 3200, -1764, 2230, 10104, -6798, -4560, -7536

Offset

-1,2

Comments

McKay-Thompson series of class 7B for the Monster group with a(0) = -4.

References

N. Elkies, The Klein quartic in number theory, pp. 51-101 of S. Levy, ed., The Eightfold Way, Cambridge Univ. Press, 1999. MR1722413 (2001a:11103)

Links

Seiichi Manyama, Table of n, a(n) for n = -1..10000

J. H. Conway and S. P. Norton, Monstrous Moonshine, Bull. Lond. Math. Soc. 11 (1979) 308-339.

N. Elkies, The Klein quartic in number theory

N. D. Elkies, Elliptic and modular curves over finite fields and related computational issues, in AMS/IP Studies in Advanced Math., 7 (1998), 21-76, esp. p. 66.

D. Ford, J. McKay and S. P. Norton, More on replicable functions, Commun. Algebra 22, No. 13, 5175-5193 (1994).

J. McKay and H. Strauss, The q-series of monstrous moonshine and the decomposition of the head characters, Comm. Algebra 18 (1990), no. 1, 253-278.

Index entries for McKay-Thompson series for Monster simple group

Formula

Euler transform of period 7 sequence [ -4, -4, -4, -4, -4, -4, 0, ...]. - Michael Somos, Mar 15 2004

G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = (u-v)^2 * (u+v) - u*v * (u+7) * (v+7). - Michael Somos, Feb 19 2007

a(n) = A052240(n) unless n = 0.

a(-1) = 1, a(n) = -(4/(n+1))*Sum_{k=1..n+1} A113957(k)*a(n-k) for n > -1. - Seiichi Manyama, Mar 29 2017

a(n) = A108481(n) - A305443(n) - A262933(n) for n > 0. - Seiichi Manyama, Oct 14 2018

Example

1/q - 4 + 2*q + 8*q^2 - 5*q^3 - 4*q^4 - 10*q^5 + 12*q^6 - 7*q^7 + 8*q^8 + ...

Mathematica

QP = QPochhammer; s = (QP[q]/QP[q^7])^4 + O[q]^50; CoefficientList[s, q] (* Jean-François Alcover, Nov 14 2015 *)

Prog

(PARI) {a(n) = local(A); if( n<-1, 0, n++; A = x * O(x^n); polcoeff( (eta(x + A) / eta(x^7 + A))^4, n))} /* Michael Somos, Feb 19 2007 */

Crossrefs

Cf. A052240, A108481, A262933, A305443.

Sequence in context: A191536 A187076 A000727 * A021879 A020806 A030210

Adjacent sequences: A030178 A030179 A030180 * A030182 A030183 A030184

Keyword

sign

Author

N. J. A. Sloane

Status

approved