|
%I #12 Mar 12 2021 22:24:44
%S 1,1,1,0,0,-1,-1,0,1,0,-1,0,0,1,2,1,-2,-3,-1,1,2,3,0,-3,-1,2,2,0,-2,
%T -6,-3,4,7,3,-2,-5,-6,2,8,3,-5,-6,-2,4,12,7,-10,-15,-6,5,13,12,-4,-18,
%U -7,11,14,6,-10,-24,-14,20,32,12,-12,-29,-24,9,36,15,-22,-30,-13,22,50,27,-36,-63,-26,24,56,45,-22,-69,-30,42,62
%N Expansion of chi(-q^3)^2 * chi(-q^5)^2 / (chi(-q) * chi(-q^15)) in powers of q where chi() is a Ramanujan theta function.
%C Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H G. C. Greubel, <a href="/A131796/b131796.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 Euler transform of period 30 sequence [ 1, 0, -1, 0, -1, 0, 1, 0, -1, 0, 1, 0, 1, 0, -2, 0, 1, 0, 1, 0, -1, 0, 1, 0, -1, 0, -1, 0, 1, 0, ...].
%F G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = v^2 + u*(2 - 4*v + u*v).
%F G.f.: Product_{k>0} (1 + x^k) * (1 + x^(15*k)) / ((1 + x^(3*k)) * (1 + x^(5*k)))^2.
%F a(n) = A131794(n) = -A131797(n) unless n=0.
%e G.f. = 1 + q + q^2 - q^5 - q^6 + q^8 - q^10 + q^13 + 2*q^14 + q^15 - 2*q^16 - ...
%t a[ n_] := SeriesCoefficient[ (QPochhammer[ q^3, q^6] QPochhammer[ q^5, q^10])^2 / (QPochhammer[ q, q^2] QPochhammer[ q^15, q^30]), {q, 0, n}]; (* _Michael Somos_, Apr 26 2015 *)
%o (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A) * eta(x^30 + A) / (eta(x + A) * eta(x^15 + A)) * (eta(x^3 + A) * eta(x^5 + A) / (eta(x^6 + A) * eta(x^10 + A)))^2, n))};
%Y Cf. A131794, A131797.
%K sign
%O 0,15
%A _Michael Somos_, Jul 16 2006
|
