A058599 McKay-Thompson series of class 27A for the Monster group.

1, 0, 3, 5, 9, 12, 20, 27, 42, 57, 81, 108, 150, 198, 267, 346, 459, 588, 765, 972, 1248, 1570, 1989, 2484, 3117, 3861, 4800, 5908, 7290, 8916, 10922, 13284, 16170, 19565, 23679, 28512, 34331, 41148, 49308, 58854, 70218, 83484, 99193, 117504, 139092

Offset

-1,3

Comments

Also McKay-Thompson series of class 27B for the Monster group.

Links

Vaclav Kotesovec, Table of n, a(n) for n = -1..10000 (terms -1..1000 from G. C. Greubel)

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

Index entries for McKay-Thompson series for Monster simple group

Formula

Expansion of T9B(q)/(1 - 3*T9B(q)/(6 + T9B(q^3))), where T9B(q) = A058091 and T9B(q^3) = T9B(q -> q^3), in powers of q. - G. C. Greubel, Jun 22 2018

a(n) ~ exp(4*Pi*sqrt(n/3)/3) / (sqrt(2) * 3^(3/4) * n^(3/4)). - Vaclav Kotesovec, Jun 28 2018

Example

T27A = 1/q +3*q +5*q^2 +9*q^3 +12*q^4 +20*q^5 +27*q^6 +42*q^7 +...

Mathematica

eta[q_]:= q^(1/24)*QPochhammer[q]; nmax := 100;  B:= (eta[q^6]/eta[q^3])*(eta[q^9]/eta[q^18])^3;  T9B := B + 4/(B)^2; A:= T9B/(6 + (T9B/.{q -> q^3})) ;  a:= CoefficientList[Series[q*T9B/(1 - 3*A + O[q]^nmax), {q, 0, 60}], q]; Table[a[[n]], {n, 1, 50}] (* G. C. Greubel, Jun 22 2018 *)

Prog

(PARI) q='q+O('q^50); B = (eta(q^6)/eta(q^3))*(eta(q^9)/eta(q^18))^3/q; B3 = (eta(q^18)/eta(q^9))*(eta(q^27)/eta(q^54))^3/q^3; T9B = B + 4/B^2; T9B3 = B3 + 4/(B3)^2; A = T9B/(6 + T9B3); Vec(T9B/(1 - 3*A)) \\ G. C. Greubel, Jun 22 2018

Crossrefs

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

Sequence in context: A190310 A046746 A344715 * A238662 A365272 A059093

Adjacent sequences: A058596 A058597 A058598 * A058600 A058601 A058602

Keyword

nonn

Author

N. J. A. Sloane, Nov 27 2000

Status

approved