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A262157 Expansion of psi(x^3)^3 / (psi(x)^2 * psi(x^2)) in powers of x where psi() is a Ramanujan theta function.

%I A262157 #14 Feb 16 2025 08:33:27
%S A262157 1,-2,2,-1,3,-8,9,-7,13,-26,29,-23,38,-72,79,-67,103,-178,196,-170,
%T A262157 248,-409,447,-403,564,-883,966,-886,1204,-1819,1984,-1861,2465,-3600,
%U A262157 3926,-3733,4846,-6893,7507,-7243,9238,-12822,13961,-13609,17104,-23263,25309
%N A262157 Expansion of psi(x^3)^3 / (psi(x)^2 * psi(x^2)) in powers of x where psi() is a Ramanujan theta function.
%C A262157 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A262157 G. C. Greubel, <a href="/A262157/b262157.txt">Table of n, a(n) for n = 0..2500</a>
%H A262157 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A262157 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A262157 Expansion of q^(-15/24) * eta(q)^2 * eta(q^6)^6 / (eta(q^2)^3 * eta(q^3)^3 * eta(q^4)^2) in powers of q.
%F A262157 Euler transform of period 12 sequence [-2, 1, 1, 3, -2, -2, -2, 3, 1, 1, -2, 0, ...].
%e A262157 G.f. = 1 - 2*x + 2*x^2 - x^3 + 3*x^4 - 8*x^5 + 9*x^6 - 7*x^7 + ...
%e A262157 G.f. = q^5 - 2*q^13 + 2*q^21 - q^29 + 3*q^37 - 8*q^45 + 9*q^53 + ...
%t A262157 a[ n_] := SeriesCoefficient[ x^(-5/8) EllipticTheta[ 2 , 0, x^(3/2)]^3 / (EllipticTheta[ 2 , 0, x^(1/2)]^2 EllipticTheta[ 2 , 0, x]), {x, 0, n}];
%o A262157 (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^2 * eta(x^6 + A)^6 / (eta(x^2 + A)^3 * eta(x^3 + A)^3 * eta(x^4 + A)^2), n))};
%o A262157 (PARI) q='q+O('q^99); Vec(eta(q)^2*eta(q^6)^6/(eta(q^2)^3*eta(q^3)^3*eta(q^4)^2)) \\ _Altug Alkan_, Jul 31 2018
%K A262157 sign
%O A262157 0,2
%A A262157 _Michael Somos_, Sep 13 2015