|
%I #6 Mar 12 2021 22:24:47
%S 1,1,0,0,-1,0,0,0,0,-2,0,0,-1,-1,0,0,0,0,0,0,0,2,0,0,-1,0,0,0,2,0,0,0,
%T 0,2,0,0,2,-1,0,0,1,0,0,0,0,0,0,0,-2,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,
%U -2,0,0,0,0,-2,0,0,0,-1,0,0,0,0,0,0,0,-2,0
%N Expansion of psi(x) * psi(-x^3) in powers of x where psi() is a Ramanujan theta function.
%C Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F Expansion of q^(-1/2) * eta(q^2)^2 * eta(q^3) * eta(q^12) / (eta(q) * eta(q^6)) in powers of q.
%F Euler transform of period 12 sequence [ 1, -1, 0, -1, 1, -1, 1, -1, 0, -1, 1, -2, ...].
%F a(3*n + 2) = a(4*n + 2) = a(4*n + 3) = 0.
%e 1 + x - x^4 - 2*x^9 - x^12 - x^13 + 2*x^21 - x^24 + 2*x^28 + 2*x^33 + ...
%e q + q^3 - q^9 - 2*q^19 - q^25 - q^27 + 2*q^43 - q^49 + 2*q^57 + 2*q^67 + ...
%t a[ n_] := SeriesCoefficient[ EllipticTheta[ 2, 0, q] EllipticTheta[ 2, Pi/4, q^3]/8^(1/2), {q, 0, 2 n + 1}]
%o (PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^2 * eta(x^3 + A) * eta(x^12 + A) / (eta(x + A) * eta(x^6 + A)), n))}
%K sign
%O 0,10
%A _Michael Somos_, Jun 04 2013
|
