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A112199 McKay-Thompson series of class 57A for the Monster group. 2
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 (list; graph; refs; listen; history; text; internal format)
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-Fran├žois 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: A224980 A154392 A069745 * A145815 A059851 A047993

Adjacent sequences:  A112196 A112197 A112198 * A112200 A112201 A112202

KEYWORD

nonn

AUTHOR

Michael Somos, Aug 28 2005

STATUS

approved

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Last modified January 17 23:15 EST 2019. Contains 319251 sequences. (Running on oeis4.)