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A058599 McKay-Thompson series of class 27A for the Monster group. 1

%I

%S 1,0,3,5,9,12,20,27,42,57,81,108,150,198,267,346,459,588,765,972,1248,

%T 1570,1989,2484,3117,3861,4800,5908,7290,8916,10922,13284,16170,19565,

%U 23679,28512,34331,41148,49308,58854,70218,83484,99193,117504,139092

%N McKay-Thompson series of class 27A for the Monster group.

%C Also McKay-Thompson series of class 27B for the Monster group.

%H Vaclav Kotesovec, <a href="/A058599/b058599.txt">Table of n, a(n) for n = -1..10000</a> (terms -1..1000 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>, Comm. Algebra 22, No. 13, 5175-5193 (1994).

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

%F Expansion of T9B(q)/(1 - 3*T9B(q)/(6 + T9B(q^3))), where T9B(q) = A058091 and T9B(q^3) = T9B(q -> q^3), in powers of q. - _G. C. Greubel_, Jun 22 2018

%F a(n) ~ exp(4*Pi*sqrt(n/3)/3) / (sqrt(2) * 3^(3/4) * n^(3/4)). - _Vaclav Kotesovec_, Jun 28 2018

%e T27A = 1/q +3*q +5*q^2 +9*q^3 +12*q^4 +20*q^5 +27*q^6 +42*q^7 +...

%t eta[q_]:= q^(1/24)*QPochhammer[q]; nmax := 100; B:= (eta[q^6]/eta[q^3])*(eta[q^9]/eta[q^18])^3; T9B := B + 4/(B)^2; A:= T9B/(6 + (T9B/.{q -> q^3})) ; a:= CoefficientList[Series[q*T9B/(1 - 3*A + O[q]^nmax), {q, 0, 60}], q]; Table[a[[n]], {n, 1, 50}] (* _G. C. Greubel_, Jun 22 2018 *)

%o (PARI) q='q+O('q^50); B = (eta(q^6)/eta(q^3))*(eta(q^9)/eta(q^18))^3/q; B3 = (eta(q^18)/eta(q^9))*(eta(q^27)/eta(q^54))^3/q^3; T9B = B + 4/B^2; T9B3 = B3 + 4/(B3)^2; A = T9B/(6 + T9B3); Vec(T9B/(1 - 3*A)) \\ _G. C. Greubel_, Jun 22 2018

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

%K nonn

%O -1,3

%A _N. J. A. Sloane_, Nov 27 2000

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Last modified February 22 15:52 EST 2019. Contains 320399 sequences. (Running on oeis4.)