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A328800 Expansion of chi(-x) * chi(x^3) in powers of x where chi() is a Ramanujan theta function.

%I A328800 #8 Feb 16 2025 08:33:58
%S A328800 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,
%T A328800 0,2,-2,0,0,3,-3,0,0,3,-3,0,0,3,-3,0,0,5,-5,0,0,4,-5,0,0,6,-5,0,0,7,
%U A328800 -7,0,0,7,-8,0,0,8,-8,0,0,11,-11,0,0,10,-12,0
%N A328800 Expansion of chi(-x) * chi(x^3) in powers of x where chi() is a Ramanujan theta function.
%C A328800 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%C A328800 Convolution square is A328797.
%C A328800 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 A328796.
%H A328800 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A328800 Expansion of q^(1/6) * (eta(q) * eta(q^6)^2) / (eta(q^2) * eta(q^3) * eta(q^12)) in powers of q.
%F A328800 Euler transform of period 12 sequence [-1, 0, 0, 0, -1, -1, -1, 0, 0, 0, -1, 0, ...].
%F A328800 G.f.: Product_{k>=1} (1 - x^(2*k-1)) * (1 + x^(6*k-3)).
%F A328800 a(n) = (-1)^n * A328802. a(4*n) = A097242(n). a(4*n + 1) = -A328796(n). a(4*n + 2) = a(4*n + 3) = 0.
%e A328800 G.f. = 1 - x - x^5 + x^8 + x^12 - x^13 + x^16 - x^17 + x^20 + ...
%e A328800 G.f. = q^-1 - q^5 - q^29 + q^47 + q^71 - q^77 + q^95 - q^101 + ...
%t A328800 a[ n_] := SeriesCoefficient[ QPochhammer[ x, x^2] QPochhammer[ -x^3, x^6], {x, 0, n}];
%o A328800 (PARI) {a(n) = my(A); if( n < 0, 0, A = x * O(x^n); polcoeff( (eta(x + A) * eta(x^6 + A)^2) / (eta(x^2 + A) * eta(x^3 + A) * eta(x^12 + A)), n))};
%Y A328800 Cf. A097242, A328796, A328797, A328802.
%K A328800 sign
%O A328800 0,25
%A A328800 _Michael Somos_, Oct 27 2019