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%I #31 Mar 12 2021 22:24:42
%S 1,7,8,22,42,63,106,190,267,428,652,932,1367,2017,2774,3950,5539,7541,
%T 10342,14184,18889,25435,33974,44720,58952,77550,100546,130780,169273,
%U 217230,278636,356566,452544,574548,726938,914742,1149685,1441787,1798740,2242436
%N McKay-Thompson series of class 18b for the Monster group.
%C Convolution inverse is A258941. - _Vaclav Kotesovec_, Nov 07 2015
%H Vaclav Kotesovec, <a href="/A058537/b058537.txt">Table of n, a(n) for n = 0..2000</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 Michael Somos, <a href="/A007191/a007191.pdf">Emails to N. J. A. Sloane, 1993</a>
%H <a href="/index/Mat#McKay_Thompson">Index entries for McKay-Thompson series for Monster simple group</a>
%F Expansion of (27 * x * (b(x)^3 + c(x)^3)^2 / (b(x) * c(x))^3)^(1/6) in powers of x where b(), c() are cubic AGM theta functions. - _Michael Somos_, Jun 16 2012
%F Expansion of q^(1/6) * a(q) / (b(q) * c(q)/3)^(1/2) in powers of q where a(), b(), c() are cubic AGM theta functions. - _Michael Somos_, Aug 20 2012
%F Convolution square is A058092. Convolution sixth power is A030197. - _Michael Somos_, Jun 16 2012
%F a(n) ~ exp(2*Pi*sqrt(2*n)/3) / (2^(3/4) * sqrt(3) * n^(3/4)). - _Vaclav Kotesovec_, Nov 07 2015
%e 1 + 7*x + 8*x^2 + 22*x^3 + 42*x^4 + 63*x^5 + 106*x^6 + 190*x^7 + 267*x^8 + ...
%e T18b = 1/q + 7*q^5 + 8*q^11 + 22*q^17 + 42*q^23 + 63*q^29 + 106*q^35 + ...
%t CoefficientList[Series[(QPochhammer[x, x]^3 + 9*x*QPochhammer[x^9, x^9]^3) / (QPochhammer[x, x]*QPochhammer[x^3, x^3]^2), {x, 0, 50}], x] (* _Vaclav Kotesovec_, Nov 07 2015 *)
%t eta[q_]:= q^(1/24)*QPochhammer[q]; A:= q^(-1/6)*eta[q]*eta[q^3]^2/(eta[q]^3 + 9*eta[q^9]^3); CoefficientList[Series[1/A, {q, 0, 60}], q] (* _G. C. Greubel_, Jun 22 2018 *)
%o (PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); A = (eta(x^3 + A) / eta(x + A))^12; polcoeff( ((1 + 27 * x * A)^2 / A)^(1/6), n))} \\ _Michael Somos_, Jun 16 2012
%o (PARI) q='q+O('q^50); A = (eta(q)^3 + 9*q*eta(q^9)^3)/(eta(q)* eta(q^3)^2); Vec(A) \\ _G. C. Greubel_, Jun 22 2018
%Y Cf. A030197, A058092, A258941.
%K nonn
%O 0,2
%A _N. J. A. Sloane_, Nov 27 2000