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A261325 Expansion of f(x^3, x^3) * f(x, x^5) / f(x, x)^2 in powers of x where f(,) is Ramanujan's general theta function.

%I A261325 #11 Feb 16 2025 08:33:26
%S A261325 1,-3,8,-18,38,-75,140,-252,439,-744,1232,-1998,3182,-4986,7700,
%T A261325 -11736,17673,-26322,38808,-56682,82070,-117867,167996,-237744,334202,
%U A261325 -466836,648224,-895014,1229148,-1679436,2283568,-3090672,4164578,-5587941,7467464,-9940482
%N A261325 Expansion of f(x^3, x^3) * f(x, x^5) / f(x, x)^2 in powers of x where f(,) is Ramanujan's general theta function.
%C A261325 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A261325 G. C. Greubel, <a href="/A261325/b261325.txt">Table of n, a(n) for n = 0..1000</a>
%H A261325 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A261325 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A261325 Expansion of f(-x^2) * f(x^3) * f(-x^6) / f(x)^3 in powers of x where f() is a Ramanujan theta function.
%F A261325 Expansion of q^(-1/3) * eta(q)^3 * eta(q^4)^3 *  eta(q^6)^4 / (eta(q^2)^8 * eta(q^3) * eta(q^12)) in powers of q.
%F A261325 Euler transform of period 12 sequence [ -3, 5, -2, 2, -3, 2, -3, 2, -2, 5, -3, 0, ...].
%F A261325 a(n) = A187153(3*n + 1) = A213265(3*n + 1) = A233670(3*n + 1) = A233672(3*n + 1).
%F A261325 2 * a(n) = A233673(3*n + 1) = - A260215(3*n + 1). a(2*n + 1) = -3 * A233698(n).
%F A261325 a(n) ~ (-1)^n * exp(2*Pi*sqrt(n/3)) / (4*3^(5/4)*n^(3/4)). - _Vaclav Kotesovec_, Nov 16 2017
%e A261325 G.f. = 1 - 3*x + 8*x^2 - 18*x^3 + 38*x^4 - 75*x^5 + 140*x^6 - 252*x^7 + ...
%e A261325 G.f. = q - 3*q^4 + 8*q^7 - 18*q^10 + 38*q^13 - 75*q^16 + 140*q^19 + ...
%t A261325 a[ n_] := SeriesCoefficient[ QPochhammer[ x^2] QPochhammer[ -x^3] QPochhammer[ x^6] / QPochhammer[ -x]^3, {x, 0, n}];
%o A261325 (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^3 * eta(x^4 + A)^3 *  eta(x^6 + A)^4 / (eta(x^2 + A)^8 * eta(x^3 + A) * eta(x^12 + A)), n))};
%Y A261325 Cf. A187153, A213265, A233670, A233672, A233673, A233698, A260215.
%K A261325 sign
%O A261325 0,2
%A A261325 _Michael Somos_, Aug 14 2015