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A213265 Expansion of psi(q) * psi(q^2) * psi(q^6) / psi(q^3)^3 in powers of q where psi() is a Ramanujan theta function.

%I A213265 #24 Feb 16 2025 08:33:17
%S A213265 1,1,1,-1,-3,-2,3,8,5,-7,-18,-12,15,38,24,-30,-75,-46,57,140,86,-104,
%T A213265 -252,-152,183,439,262,-313,-744,-442,522,1232,725,-852,-1998,-1168,
%U A213265 1365,3182,1852,-2150,-4986,-2886,3336,7700,4436,-5106,-11736,-6736,7719
%N A213265 Expansion of psi(q) * psi(q^2) * psi(q^6) / psi(q^3)^3 in powers of q where psi() is a Ramanujan theta function.
%C A213265 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A213265 G. C. Greubel, <a href="/A213265/b213265.txt">Table of n, a(n) for n = 0..1000</a>
%H A213265 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A213265 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A213265 Expansion of 1 + c(q) * c(q^4)^2 / c(q^2)^3 in powers of q where c() is a cubic AGM theta function.
%F A213265 Expansion of eta(q^2) * eta(q^3)^3 * eta(q^4)^2 * eta(q^12)^2 / (eta(q) * eta(q^6)^7) in powers of q.
%F A213265 Euler transform of period 12 sequence [ 1, 0, -2, -2, 1, 4, 1, -2, -2, 0, 1, 0, ...].
%F A213265 a(n) = A187153(n) unless n=0.
%F A213265 Empirical: Sum_{n>=0} a(n)/exp(Pi*n) = 3/4 + (1/4)*sqrt(-9 + 6*sqrt(3)). - _Simon Plouffe_, Mar 02 2021
%e A213265 1 + q + q^2 - q^3 - 3*q^4 - 2*q^5 + 3*q^6 + 8*q^7 + 5*q^8 - 7*q^9 + ...
%t A213265 a[n_]:= SeriesCoefficient[EllipticTheta[2, 0, Sqrt[q]]*EllipticTheta[2, 0, q]*EllipticTheta[2, 0, q^3]/(EllipticTheta[2, 0, q^(3/2)]^3), {q, 0, n}]; Table[a[n], {n, 0, 50}] (* _G. C. Greubel_, Dec 04 2017 *)
%o A213265 (PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A) * eta(x^3 + A)^3 * eta(x^4 + A)^2 * eta(x^12 + A)^2 / (eta(x + A) * eta(x^6 + A)^7), n))}
%Y A213265 Cf. A187153.
%K A213265 sign
%O A213265 0,5
%A A213265 _Michael Somos_, Jun 07 2012