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A199659 Expansion of q^(1/4) * (eta(q) / eta(q^3))^3 in powers of q.

%I A199659 #25 Feb 16 2025 08:33:16
%S A199659 1,-3,0,8,-9,0,17,-27,0,46,-57,0,98,-126,0,198,-243,0,371,-465,0,692,
%T A199659 -828,0,1205,-1458,0,2082,-2463,0,3463,-4104,0,5678,-6642,0,9085,
%U A199659 -10623,0,14370,-16632,0,22273,-25758,0,34178,-39246,0,51674,-59220,0,77362
%N A199659 Expansion of q^(1/4) * (eta(q) / eta(q^3))^3 in powers of q.
%C A199659 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A199659 Seiichi Manyama, <a href="/A199659/b199659.txt">Table of n, a(n) for n = 0..10000</a>
%H A199659 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A199659 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A199659 Expansion of (f(-x) / f(-x^3))^3 in powers of x where f() is a Ramanujan theta function.
%F A199659 Euler transform of period 3 sequence [ -3, -3, 0, ...].
%F A199659 G.f. is a period 1 Fourier series which satisfies f(-1 / (48 t)) = 3^(3/2) / f(t) where q = exp(2 Pi i t).
%F A199659 G.f.: (Product_{k>0} (1 - x^k) / (1 - x^(3*k)))^3.
%F A199659 Convolution cube of A137569. Convolution square is A007262. Convolution fourth power is A030182.
%F A199659 a(3*n + 2) = 0.
%F A199659 a(0) = 1, a(n) = -(3/n)*Sum_{k=1..n} A046913(k)*a(n-k) for n > 0. - _Seiichi Manyama_, Apr 28 2017
%e A199659 1 - 3*x + 8*x^3 - 9*x^4 + 17*x^6 - 27*x^7 + 46*x^9 - 57*x^10 + 98*x^12 + ...
%e A199659 1/q - 3*q^3 + 8*q^11 - 9*q^15 + 17*q^23 - 27*q^27 + 46*q^35 - 57*q^39 + ...
%t A199659 QP = QPochhammer; s = (QP[q]/QP[q^3])^3 + O[q]^60; CoefficientList[s, q] (* _Jean-François Alcover_, Nov 25 2015 *)
%o A199659 (PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x + A) / eta(x^3 + A))^3, n))}
%Y A199659 Cf. A007262, A030182, A137569.
%K A199659 sign
%O A199659 0,2
%A A199659 _Michael Somos_, Nov 08 2011