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A261992 Expansion of psi(x) * f(-x^18)^3 / (phi(-x^3) * f(-x^3)^3) in powers of x where phi(), psi(), f() are Ramanujan theta functions.

%I A261992 #11 Feb 16 2025 08:33:27
%S A261992 1,1,0,6,5,0,25,19,0,84,61,0,248,174,0,666,455,0,1662,1112,0,3912,
%T A261992 2573,0,8774,5689,0,18894,12102,0,39289,24900,0,79248,49759,0,155612,
%U A261992 96902,0,298338,184408,0,559812,343722,0,1030224,628717,0,1862647,1130418,0
%N A261992 Expansion of psi(x) * f(-x^18)^3 / (phi(-x^3) * f(-x^3)^3) in powers of x where phi(), psi(), f() are Ramanujan theta functions.
%C A261992 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A261992 G. C. Greubel, <a href="/A261992/b261992.txt">Table of n, a(n) for n = 0..1000</a>
%H A261992 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A261992 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A261992 Expansion of q^-2 * eta(q^2)^2 * eta(q^6) * eta(q^18)^3 / (eta(q) * eta(q^3)^5) in powers of q.
%F A261992 3 * a(n) = A139214(2*n + 4). a(3*n) = A233698(n). a(3*n + 1) = A128638(n+1). a(3*n + 2) = 0.
%e A261992 G.f. = 1 + x + 6*x^3 + 5*x^4 + 25*x^6 + 19*x^7 + 84*x^9 + 61*x^10 + ...
%e A261992 G.f. = q^2 + q^3 + 6*q^5 + 5*q^6 + 25*q^8 + 19*q^9 + 84*q^11 + 61*q^12 + ...
%t A261992 a[ n_] := SeriesCoefficient[ (1/2) x^(-1/8) EllipticTheta[ 2, 0, x^(1/2)] QPochhammer[ x^18]^3 / (EllipticTheta[ 4, 0, x^3] QPochhammer[ x^3]^3), {x, 0, n}];
%o A261992 (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^2 * eta(x^6 + A) * eta(x^18 + A)^3 / (eta(x + A) * eta(x^3 + A)^5), n))};
%Y A261992 Cf. A128638, A139214, A233698.
%K A261992 nonn
%O A261992 0,4
%A A261992 _Michael Somos_, Sep 07 2015