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%I #15 Jun 17 2018 22:06:04
%S 1,2,5,8,16,26,44,66,104,152,229,324,469,652,916,1250,1716,2306,3108,
%T 4116,5464,7156,9373,12144,15725,20190,25889,32952,41881,52904,66716,
%U 83688,104785,130608,162486,201336,249006,306874,377482,462860,566513,691404
%N McKay-Thompson series of class 21D for the Monster group with a(0) = 2.
%H Seiichi Manyama, <a href="/A226015/b226015.txt">Table of n, a(n) for n = -1..10000</a>
%F Expansion of (eta(q^3) * eta(q^7) / (eta(q) * eta(q^21)))^2 in powers of q.
%F Euler transform of period 21 sequence [ 2, 2, 0, 2, 2, 0, 0, 2, 0, 2, 2, 0, 2, 0, 0, 2, 2, 0, 2, 2, 0, ...].
%F G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = u*v * (u*v + 3) - (u+v) * (u^2 + 3 * u*v + v^2).
%F G.f. is a period 1 Fourier series which satisfies f(-1 / (21 t)) = f(t) where q = exp(2 Pi i t).
%F G.f.: 1/x * (Product_{k>0} (1 - x^(3*k)) * (1 - x^(7*k)) / ((1 - x^k) * (1 - x^(21*k))))^2.
%F a(n) = A058566(n) unless n=0.
%F a(n) ~ exp(4*Pi*sqrt(n/21)) / (sqrt(2) * 21^(1/4) * n^(3/4)). - _Vaclav Kotesovec_, Sep 06 2015
%e 1/q + 2 + 5*q + 8*q^2 + 16*q^3 + 26*q^4 + 44*q^5 + 66*q^6 + 104*q^7 + ...
%t nmax = 50; CoefficientList[Series[Product[((1 - x^(3*k)) * (1 - x^(7*k)) / ((1 - x^k) * (1 - x^(21*k))))^2, {k, 1, nmax}], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Sep 06 2015 *)
%t eta[q_]:= q^(1/24)*QPochhammer[q]; a:= CoefficientList[Series[q*(eta[q^3] *eta[q^7]/(eta[q]*eta[q^21]))^2, {q, 0, 60}], q]; Table[a[[n]], {n, 1, 50}] (* _G. C. Greubel_, Jun 17 2018 *)
%o (PARI) {a(n) = local(A); if( n<-1, 0, n++; A = x * O(x^n); polcoeff( (eta(x^3 + A) * eta(x^7 + A) / (eta(x + A) * eta(x^21 + A)))^2, n))}
%Y Cf. A058566.
%K nonn
%O -1,2
%A _Michael Somos_, May 22 2013
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