A058537 McKay-Thompson series of class 18b for the Monster group.
1, 7, 8, 22, 42, 63, 106, 190, 267, 428, 652, 932, 1367, 2017, 2774, 3950, 5539, 7541, 10342, 14184, 18889, 25435, 33974, 44720, 58952, 77550, 100546, 130780, 169273, 217230, 278636, 356566, 452544, 574548, 726938, 914742, 1149685, 1441787, 1798740, 2242436
Offset: 0
Keywords
Examples
1 + 7*x + 8*x^2 + 22*x^3 + 42*x^4 + 63*x^5 + 106*x^6 + 190*x^7 + 267*x^8 + ... T18b = 1/q + 7*q^5 + 8*q^11 + 22*q^17 + 42*q^23 + 63*q^29 + 106*q^35 + ...
Links
- Vaclav Kotesovec, Table of n, a(n) for n = 0..2000
- D. Ford, J. McKay and S. P. Norton, More on replicable functions, Comm. Algebra 22, No. 13, 5175-5193 (1994).
- Michael Somos, Emails to N. J. A. Sloane, 1993
- Index entries for McKay-Thompson series for Monster simple group
Programs
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Mathematica
CoefficientList[Series[(QPochhammer[x, x]^3 + 9*x*QPochhammer[x^9, x^9]^3) / (QPochhammer[x, x]*QPochhammer[x^3, x^3]^2), {x, 0, 50}], x] (* Vaclav Kotesovec, Nov 07 2015 *) eta[q_]:= q^(1/24)*QPochhammer[q]; A:= q^(-1/6)*eta[q]*eta[q^3]^2/(eta[q]^3 + 9*eta[q^9]^3); CoefficientList[Series[1/A, {q, 0, 60}], q] (* G. C. Greubel, Jun 22 2018 *) -
PARI
{a(n) = local(A); if( n<0, 0, A = x * O(x^n); A = (eta(x^3 + A) / eta(x + A))^12; polcoeff( ((1 + 27 * x * A)^2 / A)^(1/6), n))} \\ Michael Somos, Jun 16 2012 -
PARI
q='q+O('q^50); A = (eta(q)^3 + 9*q*eta(q^9)^3)/(eta(q)* eta(q^3)^2); Vec(A) \\ G. C. Greubel, Jun 22 2018
Formula
Expansion of (27 * x * (b(x)^3 + c(x)^3)^2 / (b(x) * c(x))^3)^(1/6) in powers of x where b(), c() are cubic AGM theta functions. - Michael Somos, Jun 16 2012
Expansion of q^(1/6) * a(q) / (b(q) * c(q)/3)^(1/2) in powers of q where a(), b(), c() are cubic AGM theta functions. - Michael Somos, Aug 20 2012
a(n) ~ exp(2*Pi*sqrt(2*n)/3) / (2^(3/4) * sqrt(3) * n^(3/4)). - Vaclav Kotesovec, Nov 07 2015
Comments