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A258590 Expansion of psi(-x) * psi(-x^6)^2 / f(-x^3) in powers of x where psi(), f() are Ramanujan theta functions.

%I A258590 #14 Feb 16 2025 08:33:25
%S A258590 1,-1,0,0,-1,0,0,0,0,2,0,0,1,-1,0,0,-2,0,0,0,0,2,0,0,0,-1,0,0,-1,0,0,
%T A258590 0,0,0,0,0,3,-2,0,0,-1,0,0,0,0,2,0,0,2,-1,0,0,-1,0,0,0,0,0,0,0,0,-2,0,
%U A258590 0,-1,0,0,0,0,0,0,0,3,-1,0,0,-2,0,0,0,0,4
%N A258590 Expansion of psi(-x) * psi(-x^6)^2 / f(-x^3) in powers of x where psi(), f() are Ramanujan theta functions.
%C A258590 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A258590 G. C. Greubel, <a href="/A258590/b258590.txt">Table of n, a(n) for n = 0..1000</a>
%H A258590 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A258590 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A258590 Expansion of psi(-x) * psi(x^3) / chi(-x^12)^2 = psi(-x) * chi(x^3) * psi(x^12) in powers of x where psi(), chi() are Ramanujan theta functions.
%F A258590 Expansion of q^(-3/2) * eta(q) * eta(q^4) * eta(q^6)^2 * eta(q^24)^2 / (eta(q^2) * eta(q^3) * eta(q^12)^2) in powers of q.
%F A258590 Euler transform of period 24 sequence [ -1, 0, 0, -1, -1, -1, -1, -1, 0, 0, -1, 0, -1, 0, 0, -1, -1, -1, -1, -1, 0, 0, -1, -2, ...].
%F A258590 2 * a(n) = A263577(2*n + 3).
%e A258590 G.f. = 1 - x - x^4 + 2*x^9 + x^12 - x^13 - 2*x^16 + 2*x^21 - x^25 - x^28 + ...
%e A258590 G.f. = q^3 - q^5 - q^11 + 2*q^21 + q^27 - q^29 - 2*q^35 + 2*q^45 - q^53 + ...
%t A258590 a[ n_] := SeriesCoefficient[ 2^(-3/2) x^(-13/8) EllipticTheta[ 2, Pi/4, x^(1/2)] EllipticTheta[ 2, Pi/4, x^3]^2 / QPochhammer[ x^3], {x, 0, n}];
%t A258590 a[ n_] := SeriesCoefficient[ 2^(-3/2) x^(-13/8) EllipticTheta[ 2, Pi/4, x^(1/2)] EllipticTheta[ 2, 0, x^6] QPochhammer[ -x^3, x^6], {x, 0, n}];
%t A258590 a[ n_] := SeriesCoefficient[ 2^(-3/2) x^(-1/2) EllipticTheta[ 2, Pi/4, x^(1/2)] EllipticTheta[ 2, 0, x^(3/2)]  QPochhammer[ -x^12, x^12]^2, {x, 0, n}];
%o A258590 (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A) * eta(x^4 + A) * eta(x^6 + A)^2 * eta(x^24 + A)^2 / (eta(x^2 + A) * eta(x^3 + A) * eta(x^12 + A)^2), n))};
%Y A258590 Cf. A263577.
%K A258590 sign
%O A258590 0,10
%A A258590 _Michael Somos_, Nov 06 2015