|
%I #9 Mar 12 2021 22:24:48
%S 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,
%T 3,-4,0,0,0,0,4,-2,0,0,0,-4,1,-2,2,0,0,-2,2,-2,0,0,0,0,4,0,3,0,0,-2,2,
%U -6,2,0,0,-2,4,-2,0,0,0,0,1,-2,2,0,0,-2,2,-2
%N 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 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H G. C. Greubel, <a href="/A259668/b259668.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="http://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F 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 Euler transform of period 24 sequence [ -2, 0, 0, 0, -2, -1, -2, -1, 0, 0, -2, -2, -2, 0, 0, -1, -2, -1, -2, 0, 0, 0, -2, -2, ...].
%F a(n) = A128580(2*n) = A134177(2*n) = A115660(4*n) = A128581(4*n).
%F a(6*n + 1) = -2 * A113780(n).
%e 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 G.f. = q - 2*q^5 + q^9 - 2*q^21 + 3*q^25 - 2*q^29 + 2*q^33 - 2*q^45 + ...
%t a[ n_] := SeriesCoefficient[ QPochhammer[ x^8] QPochhammer[x^12, x^24] QPochhammer[ x^6] (QPochhammer[ x, x^6] QPochhammer[ x^5, x^6])^2, {x, 0, n}];
%t a[ n_] := SeriesCoefficient[ (EllipticTheta[2, 0, x^(3/2)] EllipticTheta[ 2, Pi/4, x^(1/2)])^2 / (2^(5/2) x^(1/4) EllipticTheta[ 4, 0, x^4] EllipticTheta[ 2, Pi/4, x^3]), {x, 0, n}];
%o (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^2 * eta(x^6 + A)^3 * eta(x^8 + A) * eta(x^12 + A) / (eta(x^2 + A)^2 * eta(x^3 + A)^2 * eta(x^24 + A)), n))};
%Y Cf. A113780, A115660, A128580, A128581, A134177.
%K sign
%O 0,2
%A _Michael Somos_, Jul 02 2015
|
