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A058690 McKay-Thompson series of class 47A for the Monster group. 1
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 (list; graph; refs; listen; history; text; internal format)
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

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: A130006 A099609 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

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Last modified October 23 16:15 EDT 2018. Contains 316529 sequences. (Running on oeis4.)