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A260057 Expansion of f(-x, -x^5)^3 / (f(x, x^5) * f(-x^2, -x^2)^2) in powers of x where f(, ) is Ramanujan's general theta function.

%I A260057 #22 Feb 16 2025 08:33:26
%S A260057 1,-4,11,-24,48,-92,170,-304,526,-884,1451,-2336,3700,-5772,8876,
%T A260057 -13472,20207,-29988,44072,-64184,92680,-132760,188758,-266512,373838,
%U A260057 -521152,722266,-995432,1364684,-1861548,2527224,-3415344,4595497,-6157700,8218050,-10925848
%N A260057 Expansion of f(-x, -x^5)^3 / (f(x, x^5) * f(-x^2, -x^2)^2) in powers of x where f(, ) is Ramanujan's general theta function.
%C A260057 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A260057 G. C. Greubel, <a href="/A260057/b260057.txt">Table of n, a(n) for n = 0..2500</a>
%H A260057 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A260057 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A260057 Expansion of f(-x) * psi(x^3)^3 / (f(x)^3 * psi(-x^3)) in powers of x where psi(), f() are Ramanujan theta functions.
%F A260057 Expansion of q^(-2/3) * eta(q)^4 * eta(q^4)^3 * eta(q^6)^7 / (eta(q^2)^9 * eta(q^3)^4 * eta(q^12)) in powers of q.
%F A260057 Euler transform of period 12 sequence [-4, 5, 0, 2, -4, 2, -4, 2, 0, 5, -4, 0, ...].
%F A260057 2 * a(n) = A260215(3*n + 2).
%F A260057 a(n) ~ (-1)^n * exp(2*Pi*sqrt(n/3)) / (4*3^(5/4)*n^(3/4)). - _Vaclav Kotesovec_, Nov 16 2017
%e A260057 G.f. = 1 - 4*x + 11*x^2 - 24*x^3 + 48*x^4 - 92*x^5 + 170*x^6 - 304*x^7 + ...
%e A260057 G.f. = q^2 - 4*q^5 + 11*q^8 - 24*q^11 + 48*q^14 - 92*q^17 + 170*q^20 + ...
%t A260057 a[ n_] := SeriesCoefficient[ 2^(-5/2) x^(-3/4) QPochhammer[ x] / QPochhammer[ -x]^3 EllipticTheta[ 2, 0, x^(3/2)]^3 / EllipticTheta[ 2, Pi/4, x^(3/2)], {x, 0, n}];
%o A260057 (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^4 * eta(x^4 + A)^3 * eta(x^6 + A)^7 / (eta(x^2 + A)^9 * eta(x^3 + A)^4 * eta(x^12 + A)), n))};
%Y A260057 Cf. A260215.
%K A260057 sign
%O A260057 0,2
%A A260057 _Michael Somos_, Nov 08 2015