A058762 - OEIS

A058762

McKay-Thompson series of class 87A for Monster.

1, 0, 0, 1, 1, 1, 2, 1, 2, 2, 2, 2, 4, 3, 4, 5, 5, 5, 8, 7, 8, 10, 10, 11, 15, 14, 16, 19, 20, 21, 27, 26, 30, 35, 36, 39, 47, 47, 52, 60, 63, 68, 80, 81, 90, 101, 106, 114, 132, 135, 148, 165, 174, 187, 212, 219, 239, 264, 279, 299, 334, 348, 377, 414, 438

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OFFSET

-1,7

COMMENTS

Also McKay-Thompson series of class 87B for Monster. - Michel Marcus, Feb 24 2014

LINKS

G. C. Greubel, Table of n, a(n) for n = -1..1000

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

Expansion of (1/4)*( -3 - T29A(q) - T29A(q^3) + sqrt((3 + T29A(q) + T29A(q^2))^2 + 8*(T29A(q)*T29A(q^3) - 3)) ) in powers of q, where T29A(q) is the g.f. of A058611. - G. C. Greubel, Jul 01 2018 [Corrected by Sean A. Irvine, Sep 16 2020]

a(n) ~ exp(4*Pi*sqrt(n/87)) / (sqrt(2) * 87^(1/4) * n^(3/4)). - Vaclav Kotesovec, Jul 02 2018

EXAMPLE

T87A = 1/q + q^2 + q^3 + q^4 + 2*q^5 + q^6 + 2*q^7 + 2*q^8 + 2*q^9 + 2*q^10 + ...

MATHEMATICA

QP := QPochhammer; nmax = 100;

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^29]*G[x] + x^6*H[x^29]*H[x];

T29A := -2 + A^2/x;

T87A := (1/4)*( -3 - T29A - (T29A/.{x -> x^3}) + ((3 + T29A + (T29A/.{x -> x^3}))^2 + 8*( T29A*(T29A /. {x -> x^3}) - 3) + O[x]^nmax)^(1/2) );

a:= CoefficientList[Series[x*T87A, {x, 0, nmax}], x];