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A224825 Expansion of psi(x) * psi(x^3)^2 in powers of x where psi() is a Ramanujan theta function.

%I A224825 #17 Feb 16 2025 08:33:19
%S A224825 1,1,0,3,2,0,4,1,0,5,3,0,5,4,0,5,1,0,7,5,0,7,4,0,9,0,0,7,6,0,6,6,0,11,
%T A224825 3,0,8,5,0,10,6,0,8,2,0,9,6,0,14,8,0,10,0,0,15,7,0,7,8,0,7,4,0,14,9,0,
%U A224825 14,6,0,16,1,0,8,11,0,13,10,0,13,0,0,12
%N A224825 Expansion of psi(x) * psi(x^3)^2 in powers of x where psi() is a Ramanujan theta function.
%C A224825 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A224825 G. C. Greubel, <a href="/A224825/b224825.txt">Table of n, a(n) for n = 0..2500</a>
%H A224825 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A224825 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A224825 Expansion of q^(-7/8) * eta(q^2)^2 * eta(q^6)^4 / (eta(q) * eta(q^3)^2) in powers of q.
%F A224825 Euler transform of period 6 sequence [1, -1, 3, -1, 1, -3, ...].
%F A224825 G.f.: (Sum_{k>0} x^(k*(k-1)/2)) * (Sum_{k>0} x^(3 * k*(k-1)/2))^2.
%F A224825 a(3*n + 2) = 0. a(n) = A033768(3*n + 1). a(3*n + 1) = A224823(n).
%e A224825 G.f. = 1 + x + 3*x^3 + 2*x^4 + 4*x^6 + x^7 + 5*x^9 + 3*x^10 + 5*x^12 + 4*x^13 + ...
%e A224825 G.f. = q^7 + q^15 + 3*q^31 + 2*q^39 + 4*q^55 + q^63 + 5*q^79 + 3*q^87 + 5*q^103 + ...
%t A224825 a[ n_] := SeriesCoefficient[ EllipticTheta[ 2, 0, q^(1/2)] EllipticTheta[ 2, 0, q^(3/2)]^2 / (8 q^(7/8)), {q, 0, n}];
%o A224825 (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^2 * eta(x^6 + A)^4 / (eta(x + A) * eta(x^3 + A)^2), n))};
%Y A224825 Cf. A033768, A224823.
%K A224825 nonn
%O A224825 0,4
%A A224825 _Michael Somos_, Jul 20 2013