A058690 McKay-Thompson series of class 47A for the Monster group.

1, 0, 1, 2, 3, 3, 5, 5, 8, 9, 12, 14, 19, 22, 28, 33, 41, 48, 60, 69, 84, 98, 117, 136, 164, 188, 222, 256, 301, 345, 404, 462, 537, 614, 709, 807, 931, 1056, 1211, 1374, 1569, 1774, 2021, 2280, 2588, 2916, 3299, 3708, 4189, 4697, 5290, 5926, 6656, 7442, 8344

Offset

-1,4

Comments

Also McKay-Thompson series of class 47B for Monster. - Michel Marcus, Feb 24 2014

G.f. is a period 1 Fourier series which satisfies f(-1 / (47 t)) = f(t) where q = exp(2 Pi i t). - Michael Somos, Sep 06 2018

Links

G. C. Greubel, Table of n, a(n) for n = -1..1500

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

a(n) ~ exp(4*Pi*sqrt(n/47)) / (sqrt(2) * 47^(1/4) * n^(3/4)). - Vaclav Kotesovec, Jul 06 2018

Expansion of (eta(q^2)^7 * eta(q^8)^2 * eta(q^94)^7 * eta(q^376)^2 - 2 * eta(q) * eta(q^2)^4 * eta(q^4)^2 * eta(q^8)^2 * eta(q^47) * eta(q^94)^4 * eta(q^188)^2 * eta(q^376)^2 - eta(q)^2* eta(q^4) * eta(q^47)^2 * eta(q^188) * (eta(q^4)^3 * eta(q^188)^3 - 2 * eta(q^2) * eta(q^8)^2 * eta(q^94)* eta(q^376)^2)^2)/ (2 * eta(q)^3 * eta(q^2)^2 * eta(q^4)^2 * eta(q^8)^2 * eta(q^47)^3 * eta(q^94)^2 * eta(q^188)^2 * eta(q^376)^2) in powers of q. - Michael Somos, Jan 24 2023

Example

T47A = 1/q + q + 2*q^2 + 3*q^3 + 3*q^4 + 5*q^5 + 5*q^6 + 8*q^7 + 9*q^8 + ...

Mathematica

eta[q_]:= q^(1/24)*QPochhammer[q]; Theta[a_, b_, c_]:= Sum[q^((a*n^2 + b*n*m + c*m^2)/2), {n, -50, 50}, {m, -50, 50}]; a:= CoefficientList[ Series[q*(Theta[2, 2, 24] - Theta[4, 2, 12])/(2*eta[q]*eta[q^47]), {q, 0, 100}], q]; Table[a[[n]], {n, 1, 80}] (* G. C. Greubel, Jul 05 2018 *)
a[ n_] := With[ {T1 = QPochhammer[ q^#] QPochhammer[ q^(47 #)] &, T2 = EllipticTheta[ 2, 0, q^#] EllipticTheta[ 2, 0, q^(47 #)] &, T3 = EllipticTheta[ 3, 0, q^#] EllipticTheta[ 3, 0, q^(47 #)] &}, SeriesCoefficient[ (T3[1] + T2[1]- T3[2] - T2[2] - T2[1/2]/2) / (2 q^2 T1[1]), {q, 0, n}]]; (* Michael Somos, Sep 07 2018 *)

Prog

(PARI) {a(n) = my(A); if( n<-1, 0, n++; A = x * O(x^n); polcoeff( Ser( Vec( qfrep([2, 1; 1, 24], n+1, 1)) - Vec(qfrep([4, 1; 1, 12], n+1, 1))) / (eta(x + A) * eta(x^47 + A)), n))}; /* Michael Somos, Sep 06 2018 */

Crossrefs

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

Sequence in context: A099609 A350865 A120249 * A290369 A087153 A240176

Adjacent sequences: A058687 A058688 A058689 * A058691 A058692 A058693

Keyword

nonn

Author

N. J. A. Sloane, Nov 27 2000

Extensions

More terms from Michel Marcus, Feb 24 2014

Status

approved