A058743 McKay-Thompson series of class 69A for Monster.

1, 0, 1, 1, 1, 1, 3, 2, 2, 4, 4, 4, 7, 6, 8, 10, 10, 11, 16, 15, 18, 22, 24, 26, 34, 34, 39, 47, 50, 56, 69, 70, 80, 93, 100, 110, 131, 137, 154, 176, 189, 208, 241, 254, 283, 320, 345, 377, 430, 456, 505, 563, 607, 661, 744, 793, 871, 964, 1039, 1129, 1257

Offset

-1,7

Comments

Also McKay-Thompson series of class 69B for Monster. - Michel Marcus, Feb 24 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

Expansion of -1 + A*B/q, where A = G(q^69)*G(q) + q^14*H(q^69)*H(q), B = G(q^23)*H(q^3) - q^4*H(q^23)*G(q^3), G() is g.f. of A003114 and H() is g.f. of A003106. - G. C. Greubel, Jun 30 2018

a(n) ~ exp(4*Pi*sqrt(n/69)) / (sqrt(2) * 69^(1/4) * n^(3/4)). - Vaclav Kotesovec, Jul 02 2018

Example

T69A = 1/q + q + q^2 + q^3 + q^4 + 3*q^5 + 2*q^6 + 2*q^7 + 4*q^8 + 4*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^69]*G[x] + x^14*H[x^69]*H[x]; B:= G[x^23]*H[x^3] - x^4*H[x^23]*G[x^3]; a:= CoefficientList[Series[-1*x + A*B, {x, 0, 60}], x]; Table[a[[n]], {n, 1, 50}] (* G. C. Greubel, Jun 30 2018 *)

Crossrefs

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

Sequence in context: A287870 A308190 A235354 * A117970 A364846 A298904

Adjacent sequences: A058740 A058741 A058742 * A058744 A058745 A058746

Keyword

nonn

Author

N. J. A. Sloane, Nov 27 2000

Extensions

More terms from Michel Marcus, Feb 24 2014

Status

approved