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A259827 Expansion of phi(x) * f(-x^12)^3 / f(-x^4) in powers of x where phi(), f() are Ramanujan theta functions.

%I A259827 #10 Feb 16 2025 08:33:26
%S A259827 1,2,0,0,3,2,0,0,4,6,0,0,4,2,0,0,4,8,0,0,7,2,0,0,8,10,0,0,4,4,0,0,5,
%T A259827 10,0,0,8,4,0,0,12,10,0,0,8,6,0,0,4,14,0,0,12,2,0,0,8,14,0,0,8,4,0,0,
%U A259827 9,18,0,0,12,6,0,0,16,14,0,0,4,4,0,0,12,12
%N A259827 Expansion of phi(x) * f(-x^12)^3 / f(-x^4) in powers of x where phi(), f() are Ramanujan theta functions.
%C A259827 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%C A259827 Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).
%H A259827 G. C. Greubel, <a href="/A259827/b259827.txt">Table of n, a(n) for n = 0..2500</a>
%H A259827 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A259827 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A259827 Expansion of phi(x) * c(x^4) / (3 * x^(4/3)) in powers of x where phi() is a Ramanujan theta function and c() is a cubic AGM theta function.
%F A259827 Expansion of q^(-4/3) * eta(q^2)^5 * eta(q^12)^3 / (eta(q)^2 * eta(q^4)^3) in powers of q.
%F A259827 Euler transform of period 12 sequence [ 2, -3, 2, 0, 2, -3, 2, 0, 2, -3, 2, -3, ...].
%F A259827 a(4*n + 2) = a(4*n + 3) = 0. a(4*n + 1) = 2 * A259655(n). 6 * a(n) = A259825(3*n + 4).
%e A259827 G.f. = 1 + 2*x + 3*x^4 + 2*x^5 + 4*x^8 + 6*x^9 + 4*x^12 + 2*x^13 + ...
%e A259827 G.f. = q^4 + 2*q^7 + 3*q^16 + 2*q^19 + 4*q^28 + 6*q^31 + 4*q^40 + ...
%t A259827 a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, x] QPochhammer[ x^12]^3 / QPochhammer[ x^4], {x, 0, n}];
%o A259827 (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^5 * eta(x^12 + A)^3 / (eta(x + A)^2 * eta(x^4 + A)^3), n))};
%Y A259827 Cf. A259655, A259825.
%K A259827 nonn
%O A259827 0,2
%A A259827 _Michael Somos_, Jul 05 2015