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A279479 Expansion of f(-x, -x^5) / f(-x^24)^2 in powers of x where f(, ) is Ramanujan's general theta function.

%I A279479 #11 Feb 16 2025 08:33:38
%S A279479 1,-1,0,0,0,-1,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,-1,0,0,2,-2,0,0,0,-2,0,0,
%T A279479 2,-1,0,0,0,0,0,0,3,0,0,0,0,-2,0,0,5,-5,0,0,0,-5,0,0,6,-2,0,0,0,0,0,0,
%U A279479 7,-1,0,0,0,-5,0,0,10,-10,0,0,0,-10,0,0,12
%N A279479 Expansion of f(-x, -x^5) / f(-x^24)^2 in  powers of x where f(, ) is Ramanujan's general theta function.
%C A279479 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A279479 G. C. Greubel, <a href="/A279479/b279479.txt">Table of n, a(n) for n = 0..5000</a>
%H A279479 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A279479 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A279479 Expansion of chi(-x) * chi(x^3) / psi(x^12) in  powers of x where chi(), psi() are Ramanujan theta functions.
%F A279479 Euler transform of period 24 sequence [ -1, 0, 0, 0, -1, -1, -1, 0, 0, 0, -1, -1, -1, 0, 0, 0, -1, -1, -1, 0, 0, 0, -1, 1, ...].
%F A279479 G.f. is a period 1 Fourier series that satisfies f(-1 / (864 t)) = 3^(-1/2) (t/i)^(-1/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A279476.
%F A279479 a(4*n) = a(4*n + 1) = a(8*n + 2) = 0. a(8*n) = A096981(n).
%e A279479 G.f. = 1 - x - x^5 + x^8 + x^16 - x^21 + 2*x^24 - 2*x^25 - 2*x^29 + ...
%e A279479 G.f. = q^-5 - q^-2 - q^10 + q^19 + q^43 - q^58 + 2*q^67 - 2*q^70 + ...
%t A279479 a[ n_] := SeriesCoefficient[ QPochhammer[ x, x^6] QPochhammer[ x^5, x^6] QPochhammer[ x^6] / QPochhammer[ x^24]^2 , {x, 0, n}];
%t A279479 a[ n_] := SeriesCoefficient[ 2 x^(3/2) QPochhammer[ x, x^2] QPochhammer[ -x^3, x^6] / EllipticTheta[ 2, 0, x^6] , {x, 0, n}];
%o A279479 (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^24 + A)^2), n))};
%Y A279479 Cf. A096981, A279476.
%K A279479 sign
%O A279479 0,25
%A A279479 _Michael Somos_, Dec 12 2016