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A263923 Expansion of psi(-x^3)^2 * f(-x^2)^3 / f(-x)^2 in powers of x where psi(), f() are Ramanujan theta functions.

%I A263923 #10 Feb 16 2025 08:33:27
%S A263923 1,2,2,2,1,2,3,4,5,2,3,4,4,4,3,4,4,4,5,4,3,8,7,6,4,4,6,4,9,6,4,4,4,8,
%T A263923 5,6,9,4,7,6,7,10,6,10,7,4,9,10,5,6,6,10,6,6,9,4,9,8,10,6,6,12,8,12,8,
%U A263923 6,10,8,13,6,6,8,12,12,6,8,10,12,11,10,7,8
%N A263923 Expansion of psi(-x^3)^2 * f(-x^2)^3 / f(-x)^2 in powers of x where psi(), f() are Ramanujan theta functions.
%C A263923 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A263923 G. C. Greubel, <a href="/A263923/b263923.txt">Table of n, a(n) for n = 0..1000</a>
%H A263923 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A263923 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A263923 Expansion of q^(-11/12) * eta(q^2)^3 * eta(q^3)^2 * eta(q^12)^2 / (eta(q)^2 * eta(q^6)^2) in powers of q.
%F A263923 Euler transform of period 12 sequence [ 2, -1, 0, -1, 2, -1, 2, -1, 0, -1, 2, -3, ...].
%F A263923 -2 * a(n) = A263527(4*n + 3).
%e A263923 G.f. = 1 + 2*x + 2*x^2 + 2*x^3 + x^4 + 2*x^5 + 3*x^6 + 4*x^7 + 5*x^8 + ...
%e A263923 G.f. = q^11 + 2*q^23 + 2*q^35 + 2*q^47 + q^59 + 2*q^71 + 3*q^83 + 4*q^95 + ...
%t A263923 a[ n_] := SeriesCoefficient[ (1/2) x^(-3/4) EllipticTheta[ 2, Pi/4, x^(3/2)]^2 QPochhammer[ x^2]^3 / QPochhammer[ x]^2, {x, 0, n}];
%o A263923 (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^3 * eta(x^3 + A)^2 * eta(x^12 + A)^2 / (eta(x + A)^2 * eta(x^6 + A)^2), n))};
%Y A263923 Cf. A263527.
%K A263923 nonn
%O A263923 0,2
%A A263923 _Michael Somos_, Oct 29 2015