A112199 McKay-Thompson series of class 57A for the Monster group.

1, 1, 1, 1, 3, 2, 4, 4, 5, 6, 8, 9, 12, 13, 16, 18, 23, 25, 31, 36, 43, 48, 57, 64, 76, 86, 99, 112, 131, 146, 169, 190, 217, 243, 278, 310, 353, 394, 446, 498, 562, 624, 704, 781, 877, 972, 1088, 1204, 1345, 1488, 1656, 1829, 2033, 2240, 2486, 2738, 3030, 3334

Offset

0,5

Links

G. C. Greubel, Table of n, a(n) for n = 0..5000

K. Bringmann and H. Swisher, On a conjecture of Koike on identities between Thompson series and Roger-Ramanujan functions, Proc. Amer. Math. Soc. 135 (2007), 2317-2326. See page 2325 (A.5)

J. H. Conway and S. P. Norton, Monstrous Moonshine, Bull. Lond. Math. Soc. 11 (1979) 308-339. See page 336.

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

Index entries for McKay-Thompson series for Monster simple group

Formula

G.f.: G(x) * G(x^19) + x^4 * H(x) * H(x^19) where G() is g.f. of A003114 and H() is g.f. of A003106.

a(n) ~ exp(4*Pi*sqrt(n/57)) / (sqrt(2)*57^(1/4)*n^(3/4)). - Vaclav Kotesovec, Sep 07 2017

Example

T57A = 1/q +q^2 +q^5 +q^8 +3*q^11 +2*q^14 +4*q^17 +4*q^20 +...

Mathematica

QP = QPochhammer; G[x_] := 1/(QP[x, x^5]*QP[x^4, x^5]); H[x_] := 1/(QP[x^2, x^5]*QP[x^3, x^5]); s = (G[x]*G[x^19] + x^4*H[x]*H[x^19]) + O[x]^60; CoefficientList[s, x] (* Jean-François Alcover, Nov 15 2015 *)

Prog

(PARI) {a(n) = local(A, A1, A2); if( n<0, 0, n = 2*n ; A = x^3 * O(x^n) ; A1 = ( eta(x + A) * eta(x^19 + A) / eta(x^2 + A) / eta(x^38 + A) )^2; A2 = -subst(A1, x, -x); polcoeff( x^4 / A1 / A2 - (A1 + A2) / 4 / x, n))} \\ Michael Somos, Jan 07 2008

Crossrefs

Cf. A136569.

Sequence in context: A368677 A154392 A069745 * A145815 A059851 A327637

Adjacent sequences: A112196 A112197 A112198 * A112200 A112201 A112202

Keyword

nonn

Author

Michael Somos, Aug 28 2005

Status

approved