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A328795 Expansion of (chi(x) * chi(-x^3))^2 in powers of x where chi() is a Ramanujan theta function.

%I A328795 #8 Feb 16 2025 08:33:58
%S A328795 1,2,1,0,0,2,2,0,2,2,1,0,2,6,2,0,3,6,4,0,4,8,4,0,7,14,7,0,6,16,10,0,
%T A328795 11,20,11,0,14,32,16,0,17,38,21,0,22,46,24,0,32,66,34,0,34,78,44,0,49,
%U A328795 96,50,0,60,130,66,0,72,154,84,0,90,186,98,0,117,244
%N A328795 Expansion of (chi(x) * chi(-x^3))^2 in powers of x where chi() is a Ramanujan theta function.
%C A328795 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%C A328795 Convolution square of A328802.
%C A328795 G.f. is a period 1 Fourier series which satisfies f(-1 / (36 t)) = 2 g(t) where q = exp(2 Pi i t) and g() is g.f. for A328789.
%H A328795 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A328795 Euler transform of period 12 sequence [2, -2, 0, 0, 2, -2, 2, 0, 0, -2, 2, 0, ...].
%F A328795 G.f.: Product_{k>=1} (1 + x^(2*k-1))^2 * (1 - x^(6*k-3))^2.
%F A328795 a(n) = (-1)^n * A328797(n). a(2*n) = A112206(n).
%F A328795 a(4*n) = A328789(n). a(4*n + 1) = 2 * A328798(n). a(4*n + 2) = A328790(n). a(4*n + 3) = 0.
%e A328795 G.f. = 1 + 2*x + x^2 + 2*x^5 + 2*x^6 + 2*x^8 + 2*x^9 + x^10 + ...
%e A328795 G.f. = q^-1 + 2*q^2 + q^5 + 2*q^14 + 2*q^17 + 2*q^23 + 2*q^26 + ...
%t A328795 a[ n_] := SeriesCoefficient[ (QPochhammer[ -x, x^2] QPochhammer[ x^3, x^6])^2, {x, 0, n}];
%o A328795 (PARI) {a(n) = my(A); if( n < 0, 0, A = x * O(x^n); polcoeff( (eta(x^2 + A)^2 * eta(x^3 + A))^2 / (eta(x+ A) * eta(x^4 + A) * eta(x^6 + A))^2, n))};
%Y A328795 Cf. A112206, A328789, A328790, A328797, A328798, A328802.
%K A328795 nonn
%O A328795 0,2
%A A328795 _Michael Somos_, Oct 28 2019