%I #14 May 04 2017 15:06:11

%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

%N Expansion of eta(q) * eta(q^15) / (eta(q^6) * eta(q^10)) in powers of q.

%H Seiichi Manyama, <a href="/A131797/b131797.txt">Table of n, a(n) for n = 0..10000</a>

%F Euler transform of period 30 sequence [ -1, -1, -1, -1, -1, 0, -1, -1, -1, 0, -1, 0, -1, -1, -2, -1, -1, 0, -1, 0, -1, -1, -1, 0, -1, -1, -1, -1, -1, 0, ...].

%F G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = v*(u^2 - v) - 2 * u * (u-1).

%F G.f. is a period 1 Fourier series which satisfies f(-1 / (30 t)) = 2 * g(t) where q = exp(2 Pi i t) and g() is the g.f. for A094022.

%F G.f.: Product_{k>0} (1 - x^k) * (1 - x^(15*k)) / ((1 - x^(6*k)) * (1 - x^(10*k))).

%F a(n) = -A131794(n) = -A131796(n) unless n=0.

%F Convolution inverse of A094023.

%F a(n) = (-1)^n * A145727(n). - _Michael Somos_, Nov 11 2015

%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] QPochhammer[ q^15] / (QPochhammer[ q^6] QPochhammer[ q^10]), {q, 0, n}]; (* _Michael Somos_, Aug 26 2015 *)

%o (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A) * eta(x^15 + A) / (eta(x^6 + A) * eta(x^10 + A)), n))};

%Y Cf. A094022, A094023, A131794, A131796, A145727.

%K sign

%O 0,15

%A _Michael Somos_, Jul 16 2007