A258277 - OEIS

%I #13 Mar 12 2021 22:24:48

%S 1,-1,0,-2,2,-1,0,0,3,0,0,-2,2,-2,0,0,1,-2,0,-2,2,-1,0,0,2,0,0,-2,4,0,

%T 0,0,2,-3,0,-2,2,0,0,0,1,0,0,-4,0,-2,0,0,4,-2,0,0,2,-2,0,0,3,0,0,-2,2,

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

%N Expansion of chi(-q) * phi(-q^3) * psi(q^3) in powers of q where chi(), 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="/A258277/b258277.txt">Table of n, a(n) for n = 0..1000</a>

%H Christian Kassel, Christophe Reutenauer, <a href="http://arxiv.org/abs/1603.06357">The Fourier expansion of eta(z)eta(2z)eta(3z)/eta(6z)</a>, arXiv:1603.06357 [math.NT], 2016.

%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) * eta(q^3) * eta(q^6) / eta(q^2) in powers of q.

%F Euler transform of period 6 sequence [ -1, 0, -2, 0, -1, -2, ...].

%F G.f.: Product_{k>0} (1 - x^(3*k)) * (1 - x^(6*k)) / (1 + x^k).

%F a(n) = (-1)^n * A122865(n).

%F a(2*n + 1) = - A122856(n). a(4*n) = A002175(n). a(4*n + 1) = - A122865(n). a(4*n + 2) = a(8*n + 7) = 0. a(8*n + 3) = -2 * A121444(n). a(8*n + 5) = - A122856(n).

%F a(n) = -A258210(3*n + 1). - _Michael Somos_, May 01 2016

%e G.f. = 1 - x - 2*x^3 + 2*x^4 - x^5 + 3*x^8 - 2*x^11 + 2*x^12 - 2*x^13 + ...

%e G.f. = q - q^4 - 2*q^10 + 2*q^13 - q^16 + 3*q^25 - 2*q^34 + 2*q^37 + ...

%t a[ n_] := SeriesCoefficient[ QPochhammer[ x^3] QPochhammer[ x^6] / QPochhammer[ -x, x], {x, 0, n}];

%o (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A) * eta(x^3 + A) * eta(x^6 + A) / eta(x^2 + A), n))};

%Y Cf. A002175, A121444, A122856, A122865, A258210.

%K sign

%O 0,4

%A _Michael Somos_, May 25 2015