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%I #15 Jun 20 2018 03:21:27
%S 1,0,2,2,1,2,3,2,4,6,6,6,9,8,13,14,15,18,23,22,29,34,35,44,52,52,65,
%T 74,80,92,110,114,134,152,164,188,215,230,266,296,321,362,412,438,503,
%U 558,602,674,755,810,912,1010,1093,1210,1346,1446,1614,1772,1922,2118
%N McKay-Thompson series of class 63a for the Monster group.
%H G. C. Greubel, <a href="/A112204/b112204.txt">Table of n, a(n) for n = 0..1000</a>
%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 a(n) ~ exp(4*Pi*sqrt(n)/(3*sqrt(7))) / (sqrt(6) * 7^(1/4) * n^(3/4)). - _Vaclav Kotesovec_, May 30 2018
%F Expansion of (q*T21A)^(1/3), where T21A = A058563. - _G. C. Greubel_, Jun 20 2018
%F Expansion of q^(1/2)*(((eta(q^3)*eta(q^7))^2 - (eta(q)*eta(q^21))^2)/( eta(q)*eta(q^3)*eta(q^7)*eta(q^21)))^(2/3) in powers of q. - _G. C. Greubel_, Jun 20 2018
%e T63a = 1/q + 2*q^5 + 2*q^8 + q^11 + 2*q^14 + 3*q^17 + 2*q^20 + 4*q^23 + ...
%t CoefficientList[Series[((QPochhammer[x^3]^2 * QPochhammer[x^7]^2 - x*QPochhammer[x]^2 * QPochhammer[x^21]^2) / (QPochhammer[x] * QPochhammer[x^3] * QPochhammer[x^7] * QPochhammer[x^21]))^(2/3), {x, 0, 100}], x] (* _Vaclav Kotesovec_, May 30 2018 *)
%t eta[q_]:= q^(1/24)*QPochhammer[q]; nmax = 120; e21B := eta[q]*eta[q^3]/( eta[q^7]*eta[q^21]); T21A := 1 + e21B + 7/e21B; a:= CoefficientList[ Series[(q*T21A + O[q]^nmax)^(1/3), {q, 0, 60}], q]; Table[a[[n]], {n, 0, 50}] (* _G. C. Greubel_, Jun 20 2018 *)
%o (PARI) q='q+O('q^50); Vec((((eta(q^3)*eta(q^7))^2 - q*(eta(q)*eta(q^21) )^2)/(eta(q)*eta(q^3)*eta(q^7)*eta(q^21)))^(2/3)) \\ _G. C. Greubel_, Jun 20 2018
%K nonn
%O 0,3
%A _Michael Somos_, Aug 28 2005
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