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A139214 Expansion of q * psi(q^2) * psi(-q^9) / (phi(-q^3) * psi(-q^3)) in powers of q where phi(), psi() are Ramanujan theta functions.

%I A139214 #14 Feb 16 2025 08:33:08
%S A139214 1,0,1,3,0,3,8,0,7,18,0,15,38,0,30,75,0,57,140,0,104,252,0,183,439,0,
%T A139214 313,744,0,522,1232,0,852,1998,0,1365,3182,0,2150,4986,0,3336,7700,0,
%U A139214 5106,11736,0,7719,17673,0,11538,26322,0,17067,38808,0,25004,56682,0
%N A139214 Expansion of q * psi(q^2) * psi(-q^9) / (phi(-q^3) * psi(-q^3)) in powers of q where phi(), psi() are Ramanujan theta functions.
%C A139214 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A139214 G. C. Greubel, <a href="/A139214/b139214.txt">Table of n, a(n) for n = 1..1000</a>
%H A139214 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A139214 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A139214 Expansion of eta(q^4)^2 * eta(q^6)^2 * eta(q^9) * eta(q^36) / (eta(q^2) * eta(q^3)^3 * eta(q^12) * eta(q^18)) in powers of q.
%F A139214 G.f. is a period 1 Fourier series which satisfies f(-1 / (36 t)) = (1/4) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A139216.
%F A139214 a(3*n + 2) = 0. 2 * a(n) = A139213(n) unless n=0.
%F A139214 a(3*n) = A187100(n). a(2*n + 4) = 3 * A261992(n). - _Michael Somos_, Sep 07 2015
%e A139214 G.f. = q + q^3 + 3*q^4 + 3*q^6 + 8*q^7 + 7*q^9 + 18*q^10 + 15*q^12 + 38*q^13 + ...
%t A139214 a[ n_] := SeriesCoefficient[(1/2) EllipticTheta[ 2, 0, q] EllipticTheta[ 2, Pi/4, q^(9/2)] / (EllipticTheta[ 4, 0, q^3] EllipticTheta[ 2, Pi/4, q^(3/2)]), {q, 0, n}]; (* _Michael Somos_, Sep 07 2015 *)
%o A139214 (PARI) {a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( eta(x^4 + A)^2 * eta(x^6 + A)^2 * eta(x^9 + A) * eta(x^36 + A) / (eta(x^2 + A) * eta(x^3 + A)^3 * eta(x^12 + A) * eta(x^18 + A)), n))};
%Y A139214 Cf. A139213, A139216, A187100, A261992.
%K A139214 nonn
%O A139214 1,4
%A A139214 _Michael Somos_, Apr 11 2008