A058724 McKay-Thompson series of class 59A for the Monster group.

1, 0, 1, 1, 2, 2, 3, 3, 4, 5, 6, 7, 10, 10, 13, 15, 18, 20, 25, 28, 34, 38, 45, 50, 60, 67, 78, 88, 102, 114, 132, 147, 169, 189, 215, 240, 274, 304, 344, 383, 432, 479, 540, 597, 670, 742, 829, 916, 1023, 1128, 1255, 1384, 1536, 1690, 1874, 2059, 2277, 2501

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

Index entries for McKay-Thompson series for Monster simple group

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

Cf. A000521, A007240, A014708, A007241, A007267, A045478, etc.

Sequence in context: A125894 A091493 A305148 * A029021 A261770 A096792

Adjacent sequences: A058721 A058722 A058723 * A058725 A058726 A058727

Keyword

nonn

Author

N. J. A. Sloane, Nov 27 2000

Extensions

More terms from Michel Marcus, Feb 24 2014

Status

approved