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A259655 Expansion of psi(x^2) * f(-x^3)^3 / f(-x) in powers of x where psi(), f() are Ramanujan theta functions.

%I A259655 #9 Feb 16 2025 08:33:26
%S A259655 1,1,3,1,4,1,5,2,5,2,5,3,7,1,7,2,9,3,7,2,6,4,11,3,8,3,10,3,8,4,9,3,14,
%T A259655 2,10,2,15,6,7,5,7,3,14,5,14,3,16,5,8,4,13,5,13,3,12,4,18,5,14,4,13,5,
%U A259655 15,4,15,5,16,7,9,6,11,7,22,3,16,3,19,7,16
%N A259655 Expansion of psi(x^2) * f(-x^3)^3 / f(-x) in powers of x where psi(), f() are Ramanujan theta functions.
%C A259655 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%C A259655 Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).
%H A259655 G. C. Greubel, <a href="/A259655/b259655.txt">Table of n, a(n) for n = 0..1000</a>
%H A259655 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A259655 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A259655 Expansion of psi(x^2) * c(x) / (3 * x^(1/3)) in powers of x where psi() is a Ramanujan theta function and c() is a cubic AGM function.
%F A259655 Expansion of f(-x^3)^3 / (chi(-x) * chi(-x^2)^2) in powers of x where chi(), f() are Ramanujan theta functions.
%F A259655 Expansion of q^(-7/12) * eta(q^3)^3 * eta(q^4)^2 / (eta(q) * eta(q^2)) in powers of q.
%F A259655 Euler transform of period 12 sequence [ 1, 2, -2, 0, 1, -1, 1, 0, -2, 2, 1, -3, ...].
%F A259655 G.f.: Product_{k>0} (1 + x^k) * (1 + x^(2*k))^2 * (1 - x^(3*k))^3.
%F A259655 a(2*n) = A185220(n). a(2*n + 1) = A181648(n).
%e A259655 G.f. = 1 + x + 3*x^2 + x^3 + 4*x^4 + x^5 + 5*x^6 + 2*x^7 + 5*x^8 + ...
%e A259655 G.f. = q^7 + q^19 + 3*q^31 + q^43 + 4*q^55 + q^67 + 5*q^79 + 2*q^91 + ...
%t A259655 a[ n_] := SeriesCoefficient[ QPochhammer[ -x, x] QPochhammer[ -x^2, x^2]^2 QPochhammer[ x^3]^3, {x, 0, n}];
%t A259655 a[ n_] := SeriesCoefficient[ QPochhammer[ x^3]^3 EllipticTheta[ 2, 0, x] / (2 x^(1/4) QPochhammer[ x]), {x, 0, n}];
%o A259655 (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^3 + A)^3 * eta(x^4 + A)^2 / (eta(x + A) * eta(x^2 + A)), n))};
%Y A259655 Cf. A181648, A185220.
%K A259655 nonn
%O A259655 0,3
%A A259655 _Michael Somos_, Jul 02 2015