OFFSET
-1,5
COMMENTS
Also McKay-Thompson series of class 59B for the Monster group. - Michel Marcus, Feb 24 2014
The Monster conjugacy classes 59A and 59B are algebraic conjugates and so yield identical McKay-Thompson series. - Michael Somos, Jul 05 2014
LINKS
Vaclav Kotesovec, Table of n, a(n) for n = -1..10000 (terms -1..2500 from G. C. Greubel)
D. Ford, J. McKay and S. P. Norton, More on replicable functions, Commun. Algebra 22, No. 13, 5175-5193 (1994).
David A. Madore, Coefficients of Moonshine (McKay-Thompson) series, The Math Forum
FORMULA
G.f. A(q) satisfies 0 = f(A(q), A(q^2)) where f(u, v) = (u - v^2) * (u^2 - v) + 2*(u^2 + v^2) + 2*u*v + 2*(u + v) + 2. - Michael Somos, Jul 05 2014
Expansion of -1 + (G(q^59)*G(q) + q^12*H(q^59)*H(q))/q in powers of q, where G() is g.f. of A003114 and H() is g.f. of A003106. - G. C. Greubel, Jun 29 2018
a(n) ~ exp(4*Pi*sqrt(n/59)) / (sqrt(2) * 59^(1/4) * n^(3/4)). - Vaclav Kotesovec, Jun 29 2018
EXAMPLE
T59A = 1/q + q + q^2 + 2*q^3 + 2*q^4 + 3*q^5 + 3*q^6 + 4*q^7 + 5*q^8 + 6*q^9 + ...
MATHEMATICA
QP := QPochhammer; f[x_, y_] := QP[-x, x*y]*QP[-y, x*y]*QP[x*y, x*y]; G[x_] := f[-x^2, -x^3]/f[-x, -x^2]; H[x_] := f[-x, -x^4]/f[-x, -x^2]; A:= G[x^59]*G[x^1] + x^12*H[x^59]*H[x^1]; a:= CoefficientList[Series[A, {x, 0, 60}], x]; Table[a[[n]], {n, 1, 50}] (* G. C. Greubel, Jun 29 2018 *)
PROG
(PARI) {a(n) = my(Q1, Q2); if( n<-1, 0, Q1 = 1 + 2*x * Ser( Vec( qfrep( [2, 1; 1, 30], n+2, 1))); Q2 = 1 + 2*x * Ser( Vec( qfrep( [6, 1; 1, 10], n+2, 1))); polcoeff( 2 / ( Q1/Q2 - 1), n))}; /* Michael Somos, Jul 05 2014 */
CROSSREFS
KEYWORD
nonn
AUTHOR
N. J. A. Sloane, Nov 27 2000
EXTENSIONS
More terms from Michel Marcus, Feb 24 2014
STATUS
approved