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A254525 Expansion of f(-x^2)^2 * f(-x, x^2) / f(x^3)^3 in powers of x where f(,) is Ramanujan's general theta function.

%I A254525 #19 Feb 16 2025 08:33:24
%S A254525 1,-1,-1,-1,0,3,4,-1,-6,-5,1,10,11,-4,-19,-17,4,31,31,-9,-50,-46,11,
%T A254525 79,77,-21,-122,-112,28,183,173,-46,-273,-249,62,396,370,-98,-573,
%U A254525 -521,130,815,751,-193,-1149,-1041,261,1599,1461,-373,-2214,-1998,498,3031
%N A254525 Expansion of f(-x^2)^2 * f(-x, x^2) / f(x^3)^3 in powers of x where f(,) is Ramanujan's general theta function.
%C A254525 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A254525 G. C. Greubel, <a href="/A254525/b254525.txt">Table of n, a(n) for n = 0..1000</a>
%H A254525 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A254525 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A254525 Expansion of (chi(x) / chi(x^3)^3) * (psi(-x) / psi(-x^3))^2 in powers of x where chi(), psi() are Ramanujan theta functions.
%F A254525 Expansion of q^(1/6) * eta(q) * eta(q^3) * eta(q^4) * eta(q^12) / eta(q^6)^4 in powers of q.
%F A254525 Euler transform of period 12 sequence [ -1, -1, -2, -2, -1, 2, -1, -2, -2, -1, -1, 0, ...].
%F A254525 G.f. is a period 1 Fourier series which satisfies f(-1 / (144 t)) = 9^(1/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A254346.
%F A254525 Convolution of A062243 and A128111.
%F A254525 a(n) = (-1)^n * A132179(n).
%F A254525 a(2*n) = A230256(n). a(2*n + 1) = - A233037(n).
%e A254525 G.f. = 1 - x - x^2 - x^3 + 3*x^5 + 4*x^6 - x^7 - 6*x^8 - 5*x^9 + x^10 + ...
%e A254525 G.f. = 1/q - q^5 - q^11 - q^17 + 3*q^29 + 4*q^35 - q^41 - 6*q^47 - 5*q^53 + ...
%t A254525 eta[q_] := q^(1/24)*QPochhammer[q]; A254525[n_] := SeriesCoefficient[
%t A254525    q^(1/6)*eta[q]*eta[q^3]*eta[q^4]*eta[q^12]/eta[q^6]^4, {q, 0, n}]; Table[A254525[n], {n,0,50}] (* _G. C. Greubel_, Aug 10 2017 *)
%o A254525 (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A) * eta(x^3 + A) * eta(x^4 + A) * eta(x^12 + A) / eta(x^6 + A)^4, n))};
%Y A254525 Cf. A062243, A128111, A132179, A230256, A233037, A254346.
%K A254525 sign
%O A254525 0,6
%A A254525 _Michael Somos_, Jan 31 2015