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A261118 Expansion of psi(x)^2 * psi(-x^3)^2 / (phi(-x^4) * psi(-x^6)) in power of x where phi(), psi() are Ramanujan theta functions.

%I A261118 #10 Feb 16 2025 08:33:26
%S A261118 1,2,1,0,0,2,3,2,2,0,0,2,3,2,0,0,0,0,2,4,1,0,0,2,2,2,4,0,0,0,3,4,0,0,
%T A261118 0,0,4,2,0,0,0,4,1,2,2,0,0,2,2,2,0,0,0,0,4,0,3,0,0,2,2,6,2,0,0,2,4,2,
%U A261118 0,0,0,0,1,2,2,0,0,2,2,2,2,0,0,0,2,4,0
%N A261118 Expansion of psi(x)^2 * psi(-x^3)^2 / (phi(-x^4) * psi(-x^6)) in power of x where phi(), psi() are Ramanujan theta functions.
%C A261118 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A261118 G. C. Greubel, <a href="/A261118/b261118.txt">Table of n, a(n) for n = 0..1000</a>
%H A261118 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A261118 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A261118 Expansion of f(-x^8) * f(x, x^5)^2 / psi(-x^6) in powers of x where psi(), f() are Ramanujan theta functions.
%F A261118 Expansion of q^(-1/4) * eta(q^2)^4 * eta(q^3)^2 * eta(q^8) * eta(q^12)^3 / (eta(q)^2 * eta(q^4)^2 * eta(q^6)^3 * eta(q^24)) in powers of q.
%F A261118 Euler transform of period 24 sequence [ 2, -2, 0, 0, 2, -1, 2, -1, 0, -2, 2, -2, 2, -2, 0, -1, 2, -1, 2, 0, 0, -2, 2, -2, ...].
%F A261118 a(n) = (-1)^n * A259668(n) = A129402(2*n) = A190615(2*n) = A192013(4*n) = A000377(4*n + 1) = A129402(6*n + 1).
%F A261118 a(2*n) = A260308(n). a(2*n + 1) = 2 * A259895(n).
%e A261118 G.f. = 1 + 2*x + x^2 + 2*x^5 + 3*x^6 + 2*x^7 + 2*x^8 + 2*x^11 + 3*x^12 + ...
%e A261118 G.f. = q + 2*q^5 + q^9 + 2*q^21 + 3*q^25 + 2*q^29 + 2*q^33 + 2*q^45 + ...
%t A261118 a[n_]:= SeriesCoefficient[(-1)^(-1/8)*q^(-1/4)*(EllipticTheta[2, 0, Sqrt[q]]*EllipticTheta[2, 0, I*Sqrt[q^3]])^2/(8*EllipticTheta[3, 0, -q^4]*EllipticTheta[2, 0, I*q^3]), {q, 0, n}]; Table[a[n], {n, 0, 50}] (* _G. C. Greubel_, Jan 04 2018 *)
%o A261118 (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^4 * eta(x^3 + A)^2 * eta(x^8 + A) * eta(x^12 + A)^3 / (eta(x + A)^2 * eta(x^4 + A)^2 * eta(x^6 + A)^3 * eta(x^24 + A)), n))};
%Y A261118 Cf. A129402, A190615, A192013, A259668, A259895, A260308.
%K A261118 nonn
%O A261118 0,2
%A A261118 _Michael Somos_, Aug 08 2015