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A182036 Expansion of q^(1/2) * eta(q^2)^2 * eta(q^6)^2 / (eta(q) * eta(q^9)^3) in powers of q.

%I A182036 #17 Feb 16 2025 08:33:13
%S A182036 1,1,0,1,0,0,-1,-2,0,1,4,0,0,-1,0,-3,-8,0,4,14,0,1,-4,0,-6,-23,0,5,40,
%T A182036 0,1,-10,0,-10,-60,0,11,98,0,4,-24,0,-19,-140,0,17,224,0,4,-54,0,-31,
%U A182036 -304,0,31,478,0,9,-112,0,-50,-627,0,46,968,0,11,-224
%N A182036 Expansion of q^(1/2) * eta(q^2)^2 * eta(q^6)^2 / (eta(q) * eta(q^9)^3) in powers of q.
%C A182036 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A182036 G. C. Greubel, <a href="/A182036/b182036.txt">Table of n, a(n) for n = 0..1000</a>
%H A182036 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A182036 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A182036 Euler transform of period 18 sequence [1, -1, 1, -1, 1, -3, 1, -1, 4, -1, 1, -3, 1, -1, 1, -1, 1, 0, ...].
%F A182036 a(3*n) = A132179(n). a(3*n + 2) = 0.
%F A182036 Expansion of psi(x) * f(-x^6)^2 / f(-x^9)^3 in powers of x where psi(), f() are Ramanujan theta functions. - _Michael Somos_, Aug 10 2017
%e A182036 G.f. = 1 + x + x^3 - x^6 - 2*x^7 + x^9 + 4*x^10 - x^13 - 3*x^15 - 8*x^16 + ...
%e A182036 G.f. = 1/q + q + q^5 - q^11 - 2*q^13 + q^17 + 4*q^19 - q^25 - 3*q^29 - 8*q^31 + ...
%t A182036 eta[x_] := x^(1/24)*QPochhammer[x]; A182036[n_] := SeriesCoefficient[q^(1/2)*(eta[q^2]* eta[q^6])^2/(eta[q]*eta[q^9]^3), {q,0,n}]; Table[A182036[n], {n,0,50}] (* _G. C. Greubel_, Aug 09 2017 *)
%t A182036 a[ n_] := SeriesCoefficient[ 1/2 x^(-1/8) EllipticTheta[ 2, 0, x^(1/2)] QPochhammer[ x^6]^2 / QPochhammer[ x^9]^3, {x, 0, n}]; (* _Michael Somos_, Aug 10 2017 *)
%o A182036 (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^2 * eta(x^6 + A)^2 / (eta(x + A) * eta(x^9 + A)^3), n))};
%Y A182036 Cf. A132179.
%K A182036 sign
%O A182036 0,8
%A A182036 _Michael Somos_, Apr 07 2012