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.
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
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
KEYWORD
sign
AUTHOR
STATUS
approved