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A328802 Expansion of chi(x) * chi(-x^3) in powers of x where chi() is a Ramanujan theta function.

%I A328802 #8 Feb 16 2025 08:33:58
%S A328802 1,1,0,0,0,1,0,0,1,0,0,0,1,1,0,0,1,1,0,0,1,1,0,0,2,2,0,0,1,2,0,0,2,2,
%T A328802 0,0,3,3,0,0,3,3,0,0,3,3,0,0,5,5,0,0,4,5,0,0,6,5,0,0,7,7,0,0,7,8,0,0,
%U A328802 8,8,0,0,11,11,0,0,10,12,0,0,13,12,0,0,15
%N A328802 Expansion of chi(x) * chi(-x^3) in powers of x where chi() is a Ramanujan theta function.
%C A328802 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%C A328802 Convolution square is A328795.
%C A328802 G.f. is a period 1 Fourier series which satisfies f(-1 / (1728 t)) = 2^(1/2) g(t) where q = exp(2 Pi i t) and g() is g.f. for A097242.
%H A328802 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A328802 Expansion of q^(1/6) * (eta(q^2)^2 * eta(q^3)) / (eta(q) * eta(q^4) * eta(q^6)) in powers of q.
%F A328802 Euler transform of period 12 sequence [1, -1, 0, 0, 1, -1, 1, 0, 0, -1, 1, 0, ...].
%F A328802 G.f.: Product_{k>=1} (1 + x^(2*k-1)) * (1 - x^(6*k-3)).
%F A328802 a(n) = (-1)^n * A328800. a(4*n) = A097242(n). a(4*n + 1) = A328796(n). a(4*n + 2) = a(4*n + 3) = 0.
%e A328802 G.f. = 1 + x + x^5 + x^8 + x^12 + x^13 + x^16 + x^17 + x^20 + ...
%e A328802 G.f. = q^-1 + q^5 + q^29 + q^47 + q^71 + q^77 + q^95 + q^101 + ...
%t A328802 a[ n_] := SeriesCoefficient[ QPochhammer[ -x, x^2] QPochhammer[ x^3, x^6], {x, 0, n}];
%o A328802 (PARI) {a(n) = my(A); if( n < 0, 0, A = x * O(x^n); polcoeff( (eta(x^2 + A)^2 * eta(x^3 + A)) / (eta(x + A) * eta(x^4 + A) * eta(x^6 + A)), n))};
%Y A328802 Cf. A097242, A328795, A328796, A328800.
%K A328802 nonn
%O A328802 0,25
%A A328802 _Michael Somos_, Oct 28 2019