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%I #27 Jun 29 2018 10:18:40
%S 1,0,1,1,2,2,3,3,4,5,6,7,10,10,13,15,18,20,25,28,34,38,45,50,60,67,78,
%T 88,102,114,132,147,169,189,215,240,274,304,344,383,432,479,540,597,
%U 670,742,829,916,1023,1128,1255,1384,1536,1690,1874,2059,2277,2501
%N McKay-Thompson series of class 59A for the Monster group.
%C Also McKay-Thompson series of class 59B for the Monster group. - _Michel Marcus_, Feb 24 2014
%C The Monster conjugacy classes 59A and 59B are algebraic conjugates and so yield identical McKay-Thompson series. - _Michael Somos_, Jul 05 2014
%H Vaclav Kotesovec, <a href="/A058724/b058724.txt">Table of n, a(n) for n = -1..10000</a> (terms -1..2500 from G. C. Greubel)
%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 G.f. A(q) satisfies 0 = f(A(q), A(q^2)) where f(u, v) = (u - v^2) * (u^2 - v) + 2*(u^2 + v^2) + 2*u*v + 2*(u + v) + 2. - _Michael Somos_, Jul 05 2014
%F Expansion of -1 + (G(q^59)*G(q) + q^12*H(q^59)*H(q))/q in powers of q, where G() is g.f. of A003114 and H() is g.f. of A003106. - _G. C. Greubel_, Jun 29 2018
%F a(n) ~ exp(4*Pi*sqrt(n/59)) / (sqrt(2) * 59^(1/4) * n^(3/4)). - _Vaclav Kotesovec_, Jun 29 2018
%e T59A = 1/q + q + q^2 + 2*q^3 + 2*q^4 + 3*q^5 + 3*q^6 + 4*q^7 + 5*q^8 + 6*q^9 + ...
%t QP := QPochhammer; f[x_, y_] := QP[-x, x*y]*QP[-y, x*y]*QP[x*y, x*y]; G[x_] := f[-x^2, -x^3]/f[-x, -x^2]; H[x_] := f[-x, -x^4]/f[-x, -x^2]; A:= G[x^59]*G[x^1] + x^12*H[x^59]*H[x^1]; a:= CoefficientList[Series[A, {x, 0, 60}], x]; Table[a[[n]], {n, 1, 50}] (* _G. C. Greubel_, Jun 29 2018 *)
%o (PARI) {a(n) = my(Q1, Q2); if( n<-1, 0, Q1 = 1 + 2*x * Ser( Vec( qfrep( [2, 1; 1, 30], n+2, 1))); Q2 = 1 + 2*x * Ser( Vec( qfrep( [6, 1; 1, 10], n+2, 1))); polcoeff( 2 / ( Q1/Q2 - 1), n))}; /* _Michael Somos_, Jul 05 2014 */
%Y Cf. A000521, A007240, A014708, A007241, A007267, A045478, etc.
%K nonn
%O -1,5
%A _N. J. A. Sloane_, Nov 27 2000
%E More terms from _Michel Marcus_, Feb 24 2014
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