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%I #17 Aug 14 2018 21:05:17
%S 1,4,14,37,93,210,454,925,1824,3463,6408,11538,20353,35161,59726,
%T 99775,164337,266978,428521,679861,1067415,1659205,2555617,3902055,
%U 5909867,8881849,13252334,19637281,28909989,42297267,61520450,88976461,127996994
%N Expansion of q^(-5/24) * eta(q^3)^3 / eta(q)^4 in powers of q.
%H G. C. Greubel, <a href="/A187428/b187428.txt">Table of n, a(n) for n = 0..2500</a>
%F Euler transform of period 3 sequence [ 4, 4, 1, ...].
%F G.f. is a period 1 Fourier series which satisfies f(-1 / (576 t)) = 648^(-1/2) (t/i)^(-1/2) g(t) where q = exp(2 Pi i t) and g() is g.f. for A187427.
%F G.f.: Product_{k>0} (1 - x^(3*k))^3 / (1 - x^k)^4.
%F a(n) ~ exp(sqrt(2*n)*Pi)/(12*sqrt(3)*n). - _Vaclav Kotesovec_, Sep 07 2015
%e 1 + 4*x + 14*x^2 + 37*x^3 + 93*x^4 + 210*x^5 + 454*x^6 + 925*x^7 + ...
%e q^5 + 4*q^29 + 14*q^53 + 37*q^77 + 93*q^101 + 210*q^125 + 454*q^149 + ...
%t nmax = 40; CoefficientList[Series[Product[(1 - x^(3*k))^3 / (1 - x^k)^4, {k, 1, nmax}], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Sep 07 2015 *)
%t eta[q_] := q^(1/24)*QPochhammer[q]; CoefficientList[Series[q^(-5/24) *eta[q^3]^3/eta[q]^4, {q, 0, 50}], q] (* _G. C. Greubel_, Aug 14 2018 *)
%o (PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^3 + A)^3 / eta(x + A)^4, n))}
%Y Cf. A187427.
%K nonn
%O 0,2
%A _Michael Somos_, Mar 09 2011
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