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%I #28 Jan 25 2023 10:04:47
%S 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,
%T 188,222,256,301,345,404,462,537,614,709,807,931,1056,1211,1374,1569,
%U 1774,2021,2280,2588,2916,3299,3708,4189,4697,5290,5926,6656,7442,8344
%N McKay-Thompson series of class 47A for the Monster group.
%C Also McKay-Thompson series of class 47B for Monster. - _Michel Marcus_, Feb 24 2014
%C 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
%H G. C. Greubel, <a href="/A058690/b058690.txt">Table of n, a(n) for n = -1..1500</a>
%H D. Ford, J. McKay and S. P. Norton, <a href="http://dx.doi.org/10.1080/00927879408825127">More on replicable functions</a>, Commun. Algebra 22, No. 13, 5175-5193 (1994).
%H David A. Madore, <a href="http://mathforum.org/kb/thread.jspa?forumID=253&threadID=1602206&messageID=5836094">Coefficients of Moonshine (McKay-Thompson) series</a>, The Math Forum
%H <a href="/index/Mat#McKay_Thompson">Index entries for McKay-Thompson series for Monster simple group</a>
%F a(n) ~ exp(4*Pi*sqrt(n/47)) / (sqrt(2) * 47^(1/4) * n^(3/4)). - _Vaclav Kotesovec_, Jul 06 2018
%F 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
%e 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 + ...
%t 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 *)
%t 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 *)
%o (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 */
%Y Cf. A000521, A007240, A014708, A007241, A007267, A045478, etc.
%K nonn
%O -1,4
%A _N. J. A. Sloane_, Nov 27 2000
%E More terms from _Michel Marcus_, Feb 24 2014
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