A058206 McKay-Thompson series of class 12C for the Monster group.
1, 7, 15, 71, 106, 273, 486, 961, 1563, 3040, 4692, 8199, 12773, 20919, 31569, 50552, 74368, 114504, 167366, 250033, 358845, 527650, 745688, 1073784, 1504452, 2129317, 2947224, 4122518, 5644462, 7792122, 10585876, 14446420, 19450323, 26307536, 35131220, 47077341, 62449405, 82987854, 109317927, 144252191
Offset: 0
Keywords
Examples
T12C = 1/q + 7*q + 15*q^3 + 71*q^5 + 106*q^7 + 273*q^9 + 486*q^11 + ...
Links
- G. C. Greubel, Table of n, a(n) for n = 0..1000 (terms 0..128 from G. A. Edgar)
- D. Ford, J. McKay and S. P. Norton, More on replicable functions, Commun. 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
QP := QPochhammer; CoefficientList[Series[QP[x^2]^6*QP[x^3]^6 / (QP[x]^6*QP[x^6]^6) + x*QP[x]^6*QP[x^6]^6 / (QP[x^2]^6*QP[x^3]^6), {x, 0, 66}], x] (* Indranil Ghosh, Mar 14 2017 *) eta[q_]:= q^(1/24)*QPochhammer[q]; A := q^(1/2)*(eta[q^2]* eta[q^3]/( eta[q]*eta[q^6]))^6; a := CoefficientList[Series[A + q/A, {q, 0, 60}], q]; Table[a[[n]], {n, 1, 50}] (* G. C. Greubel, Jun 25 2018 *) a[ n_] := With[{A = (QPochhammer[ x^3, x^6] / QPochhammer[ x, x^2])^6 }, SeriesCoefficient[ A + x / A, {x, 0, n}]]; Table[ a[ n], {n, 0, 39}] (* Michael Somos, Jul 06 2018 *) -
PARI
q='q+O('q^66); Vec( eta(q^2)^6*eta(q^3)^6 / (eta(q)^6*eta(q^6)^6) + q* eta(q)^6*eta(q^6)^6 / (eta(q^2)^6*eta(q^3)^6) ) \\ Joerg Arndt, Mar 13 2017
Formula
Expansion of q^(1/2)*(eta(q^2)*eta(q^3)/(eta(q)*eta(q^6)))^6 + (eta(q)*eta(q^6)/(eta(q^2)*eta(q^3)))^6 in powers of q. - G. A. Edgar, Mar 13 2017
a(n) ~ exp(2*Pi*sqrt(n/3)) / (2*3^(1/4)*n^(3/4)). - Vaclav Kotesovec, Mar 18 2017
G.f. is a period 1 Fourier series which satisfies f(-1 / (24 t)) = f(t) where q = exp(2 Pi i t). - Michael Somos, Jul 06 2018
Extensions
More terms from G. A. Edgar, Mar 13 2017