A058708 McKay-Thompson series of class 54A for the Monster group.

1, 0, 1, 1, 3, 2, 4, 3, 6, 5, 9, 8, 14, 12, 19, 18, 27, 26, 37, 36, 52, 50, 69, 68, 93, 93, 122, 124, 162, 164, 210, 216, 274, 281, 351, 364, 451, 468, 572, 598, 726, 760, 913, 960, 1148, 1208, 1431, 1512, 1782, 1884, 2206, 2339, 2727, 2892, 3353, 3564, 4114

Offset

-1,5

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

Expansion of -1 + A*B/x in powers of q, where A = G(x^54)*G(x) + x^11* H(x^54)*H(x), B = G(x^27)*H(x^2) - x^5*H(x^27)*G(x^2), G() is g.f. of A003114 and H() is g.f. of A003106. - G. C. Greubel, Jun 29 2018

a(n) ~ exp(2*Pi*sqrt(2*n/3)/3) / (6^(3/4) * n^(3/4)). - Vaclav Kotesovec, Jun 29 2018

Example

T54A = 1/q + q + q^2 + 3*q^3 + 2*q^4 + 4*q^5 + 3*q^6 + 6*q^7 + 5*q^8 + 9*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^54]*G[x^1] + x^11*H[x^54]*H[x^1]; B:= G[x^27]*H[x^2] - x^5*H[x^27]*G[x^2]; a:= CoefficientList[Series[-x + A*B, {x, 0, 60}], x]; Table[a[[n]], {n, 1, 50}] (* G. C. Greubel, Jun 29 2018 *)

Crossrefs

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

Sequence in context: A133131 A261985 A026923 * A141731 A294145 A024856

Adjacent sequences: A058705 A058706 A058707 * A058709 A058710 A058711

Keyword

nonn

Author

N. J. A. Sloane, Nov 27 2000

Extensions

More terms from Michel Marcus, Feb 24 2014

Status

approved