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A187427 Expansion of q^(3/8) * eta(q)^3 / eta(q^3)^4 in powers of q.

%I A187427 #16 Aug 14 2018 21:05:09
%S A187427 1,-3,0,9,-12,0,27,-42,0,82,-111,0,207,-279,0,486,-630,0,1055,-1362,0,
%T A187427 2205,-2775,0,4374,-5472,0,8427,-10389,0,15696,-19224,0,28539,-34614,
%U A187427 0,50630,-61059,0,88119,-105483,0,150417,-179178,0,252727,-299325,0,418068
%N A187427 Expansion of q^(3/8) * eta(q)^3 / eta(q^3)^4 in powers of q.
%H A187427 G. C. Greubel, <a href="/A187427/b187427.txt">Table of n, a(n) for n = 0..2500</a>
%F A187427 Euler transform of period 3 sequence [ -3, -3, 1, ...].
%F A187427 G.f. is a period 1 Fourier series which satisfies f(-1 / (1728 t)) = (9/8)^(1/2) (t/i)^(-1/2) g(t) where q = exp(2 Pi i t) and g() is g.f. for A187428.
%F A187427 G.f.: Product_{k>0} (1 - x^k)^3 / (1 - x^(3*k))^4.
%F A187427 a(3*n) = A053762(n). a(3*n + 1) = -3 * A187428(n). a(3*n + 2) = 0.
%e A187427 1 - 3*x + 9*x^3 - 12*x^4 + 27*x^6 - 42*x^7 + 82*x^9 - 111*x^10 + ...
%e A187427 q^-3 - 3*q^5 + 9*q^21 - 12*q^29 + 27*q^45 - 42*q^53 + 82*q^69 - 111*q^77 + ...
%t A187427 QP = QPochhammer; s = QP[q]^3/QP[q^3]^4 + O[q]^50; CoefficientList[s, q] (* _Jean-François Alcover_, Nov 25 2015 *)
%t A187427 eta[q_]:= q^(1/24)*QPochhammer[q]; CoefficientList[Series[q^(3/8) *eta[q]^3/ eta[q^3]^4, {q, 0, 50}], q] (* _G. C. Greubel_, Aug 14 2018 *)
%o A187427 (PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^3 / eta(x^3 + A)^4, n))}
%Y A187427 Cf. A053762, A187248.
%K A187427 sign
%O A187427 0,2
%A A187427 _Michael Somos_, Mar 09 2011