%I #12 Feb 16 2025 08:33:37

%S 1,2,1,2,2,-1,1,1,-2,0,2,-1,-1,2,-2,-1,0,-2,-2,-2,0,-1,1,0,2,-2,-5,0,

%T -2,0,0,1,-2,0,0,-3,-1,0,0,-2,1,-1,2,2,0,0,0,-2,-2,0,2,2,-2,0,-2,2,1,

%U 1,0,0,0,-1,2,0,4,0,1,2,0,2,-1,0,0,2,-2,-2,4,-1

%N Expansion of psi(x)^2 * chi(-x^5) in powers of x where psi(), chi() are Ramanujan theta functions.

%C Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

%H G. C. Greubel, <a href="/A279362/b279362.txt">Table of n, a(n) for n = 0..1000</a>

%H Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>

%F Expansion of q^(-1/24) * eta(q^2)^4 * eta(q^5) / (eta(q)^2 * eta(q^10)) in powers of q.

%F Euler transform of period 10 sequence [ 2, -2, 2, -2, 1, -2, 2, -2, 2, -2, ...].

%e G.f. = 1 + 2*x + x^2 + 2*x^3 + 2*x^4 - x^5 + x^6 + x^7 - 2*x^8 + 2*x^10 + ...

%e G.f. = q + 2*q^25 + q^49 + 2*q^73 + 2*q^97 - q^121 + q^145 + q^169 - ...

%t a[ n_] := SeriesCoefficient[ x^(-1/4) / 4 EllipticTheta[ 2, 0, x^(1/2)]^2 QPochhammer[ x^5, x^10], {x, 0, n}];

%o (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^4 * eta(x^5 + A) / (eta(x + A)^2 * eta(x^10 + A)), n))};

%o (PARI) lista(nn) = {q='q+O('q^nn); Vec(eta(q^2)^4*eta(q^5)/(eta(q)^2*eta(q^10)))} \\ _Altug Alkan_, Mar 21 2018

%K sign

%O 0,2

%A _Michael Somos_, Dec 10 2016