|
%I #14 Mar 12 2021 22:24:46
%S 1,1,0,4,4,1,4,4,5,0,0,8,4,4,4,8,9,4,0,4,12,1,4,8,8,4,0,8,8,4,8,16,8,
%T 5,0,12,12,0,8,12,13,0,0,8,8,8,12,8,16,4,0,16,12,4,4,20,13,4,0,16,20,
%U 8,8,8,8,9,0,12,16,4,12,12,16,0,0,16,20,4,8
%N Expansion of chi(x) * f(x^3)^3 in powers of x where chi(), f() 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="/A213056/b213056.txt">Table of n, a(n) for n = 0..2500</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 q^(-1/3) * eta(q^2)^2 * eta(q^6)^9 / (eta(q) * eta(q^3)^3 * eta(q^4) * eta(q^12)^3) in powers of q.
%F Expansion of q^(-1/9) times theta series of cubic lattice with respect to point [0, 0, 1/3] in powers of q^(1/3).
%F Euler transform of period 12 sequence [ 1, -1, 4, 0, 1, -7, 1, 0, 4, -1, 1, -3, ...].
%F G.f.: Product_{k>0} (1 - (-x)^(3*k))^3 * (1 + x^(2*k-1)).
%F a(4*n + 1) = a(n). a(8*n + 2) = 0.
%e G.f. = 1 + x + 4*x^3 + 4*x^4 + x^5 + 4*x^6 + 4*x^7 + 5*x^8 + 8*x^11 + 4*x^12 + ...
%e G.f. = q + q^4 + 4*q^10 + 4*q^13 + q^16 + 4*q^19 + 4*q^22 + 5*q^25 + 8*q^34 + ...
%t CoefficientList[QPochhammer[q^2]^2*QPochhammer[-q^3]^3 / (QPochhammer[q] * QPochhammer[q^4]) + O[q]^80, q] (* _Jean-François Alcover_, Nov 05 2015 *)
%t eta[q_]:= q^(1/24)*QPochhammer[q]; CoefficientList[Series[q^(-1/3)* eta[q^2]^2*eta[q^6]^9/(eta[q]*eta[q^3]^3*eta[q^4]*eta[q^12]^3), {q, 0, 50}], q] (* _G. C. Greubel_, Aug 12 2018 *)
%o (PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^2 * eta(x^6 + A)^9 / (eta(x + A) * eta(x^3 + A)^3 * eta(x^4 + A) * eta(x^12 + A)^3) ,n))}
%K nonn
%O 0,4
%A _Michael Somos_, Jun 03 2012
|
