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A101558 McKay-Thompson series of class 2A for the Monster group. 6


%S 1,0,4372,96256,1240002,10698752,74428120,431529984,2206741887,

%T 10117578752,42616961892,166564106240,611800208702,2125795885056,

%U 7040425608760,22327393665024,68134255043715,200740384538624

%N McKay-Thompson series of class 2A for the Monster group.

%C Hauptmodul for Gamma_0(2)+.

%D T. Gannon, Moonshine Beyond the Monster, Cambridge, 2006; see p. 423.

%H Seiichi Manyama, <a href="/A101558/b101558.txt">Table of n, a(n) for n = -1..10000</a>

%H R. E. Borcherds, <a href="http://dx.doi.org/10.1090/S0273-0979-08-01209-3">Review of "Moonshine Beyond the Monster ..." (Cambridge, 2006)</a>, Bull. Amer. Math. Soc., 45 (2008), 675-679.

%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 N. Heninger, E. M. Rains and N. J. A. Sloane, <a href="https://arxiv.org/abs/math/0509316">On the Integrality of n-th Roots of Generating Functions</a>, arXiv:math/0509316 [math.NT], 2005-2006; J. Combinatorial Theory, Series A, 113 (2006), 1732-1745.

%H <a href="/index/Mat#McKay_Thompson">Index entries for McKay-Thompson series for Monster simple group</a>

%F a(n) ~ exp(2*Pi*sqrt(2*n)) / (2^(3/4)*n^(3/4)). - _Vaclav Kotesovec_, Apr 01 2017

%e T2A = 1/q + 4372q + 96256q^2 + 1240002q^3 + ...

%t eta[q_]:= q^(1/24)*QPochhammer[q]; f2A:= (eta[q]/eta[q^2])^24*(1 + 64*( eta[q^2]/eta[q])^24)^2; a:= CoefficientList[Series[q*(f2A - 104), {q, 0, 50}], q]; Table[a[[n]], {n,1,50}] (* _G. C. Greubel_, May 10 2018 *)

%o (PARI) {a(n) = my(A); if( n<-1, 0, A = prod(k=1, n\2+1, 1 - x^(2*k-1), 1 + x^2 * O(x^n))^24; polcoeff(64^2*x/A + A/x + 24, n))};

%Y A045478, A007241, A106207, A007267, A101558 are all essentially the same sequence.

%Y Cf. A007241 (same except for 0th term), A007267, A045478.

%K nonn

%O -1,3

%A _Michael Somos_, Dec 06 2004

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Last modified November 28 19:48 EST 2020. Contains 338726 sequences. (Running on oeis4.)