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Expansion of chi(-x) * chi(-x^4) / (chi(-x^3) * chi(-x^12)) in powers of x where chi() is a Ramanujan theta function.
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%I #8 Mar 12 2021 22:24:48

%S 1,-1,0,0,-1,0,1,-1,1,1,-1,0,1,-2,1,1,-1,0,2,-3,0,2,-2,0,4,-4,1,3,-5,

%T 0,5,-6,3,4,-6,0,6,-9,3,6,-7,1,9,-12,2,9,-11,1,15,-17,4,12,-18,2,19,

%U -23,8,17,-23,3,22,-31,9,23,-28,3,31,-41,8,31,-39,5,46

%N Expansion of chi(-x) * chi(-x^4) / (chi(-x^3) * chi(-x^12)) in powers of x where chi() is a Ramanujan theta function.

%C The sequence generating function appears on the right side of one of Ramanujan's Forty identities.

%C Rogers-Ramanujan functions: G(q) (see A003114), H(q) (A003106).

%C Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

%H G. C. Greubel, <a href="/A281743/b281743.txt">Table of n, a(n) for n = 0..1000</a>

%H Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>

%F Expansion of G(x^8) * H(x^3) - x * G(x^3) * H(x^8) in powers of x where G(), H() are Rogers-Ramanujan functions.

%F Expansion of q^(-5/12) * eta(q) * eta(q^4) * eta(q^6) * eta(q^24) / (eta(q^2) * eta(q^3) * eta(q^8) * eta(q^12)) in powers of q.

%F Euler transform of period 24 sequence [-1, 0, 0, -1, -1, 0, -1, 0, 0, 0, -1, 0, -1, 0, 0, 0, -1, 0, -1, -1, 0, 0, -1, 0, ...].

%e G.f. = 1 - x - x^4 + x^6 - x^7 + x^8 + x^9 - x^10 + x^12 - 2*x^13 + ...

%e G.f. = q^5 - q^17 - q^53 + q^77 - q^89 + q^101 + q^113 - q^125 + ...

%t a[ n_] := SeriesCoefficient[ QPochhammer[ x, x^2] QPochhammer[ x^4, x^8] QPochhammer[ -x^3, x^3] QPochhammer[ -x^12, x^12], {x, 0, n}];

%o (PARI) {a(n) = if( n<0, 0, my(A = x * O(x^n)); polcoeff( eta(x + A) * eta(x^4 + A) * eta(x^6 + A) * eta(x^24 + A) / (eta(x^2 + A) * eta(x^3 + A) * eta(x^8 + A) * eta(x^12 + A)), n))};

%K sign

%O 0,14

%A _Michael Somos_, Jan 28 2017