A145466 - OEIS

%I #16 Mar 25 2017 09:40:07

%S 1,-1,-1,0,0,2,-1,0,0,0,3,-2,-2,0,0,4,-3,-2,0,0,7,-5,-3,0,0,10,-6,-4,

%T 0,0,15,-10,-7,0,0,20,-13,-8,0,0,28,-19,-13,0,0,38,-25,-16,0,0,52,-34,

%U -23,0,0,68,-44,-28,0,0,91,-60,-40,0,0,118,-76,-48,0,0,153,-100,-66,0,0,196,-127,-82,0,0,252,-164,-107,0,0

%N Expansion of q^(1/6) * eta(q) / eta(q^5) in powers of q.

%H Seiichi Manyama, <a href="/A145466/b145466.txt">Table of n, a(n) for n = 0..10000</a>

%F Expansion of 1 / (G(x) * H(x)) = G(x^5)^2 - x * G(x^5) * H(x^5) - x^2 * H(x^5)^2 in powers of x where G(), H() are the Rogers-Ramanujan functions.

%F Euler transform of period 5 sequence [ -1, -1, -1, -1, 0, ...].

%F G.f. is a period 1 Fourier series which satisfies f(-1 / (180 t)) = 5^(1/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. of A035959.

%F Given g.f. A(x), then B(x) = A(x^3)^2 / x satisfies 0 = f(B(x), B(x^2)) where f(u, v) = u^3 + v^3 - 5*u*v - u^2*v^2.

%F Given g.f. A(x), then B(x) = A(x^3)^2 / x satisfies 0 = f(B(x), B(x^2), B(x^4)) where f(u, v, w) = v * u^2 * w^2 + 5 * u * w * (u + w) - v^2 * (u^2 + u*w + w^2).

%F a(5*n + 3) = a(5*n + 4) = 0.

%F G.f.: 1 / (Product_{k>0} P(5, x^k)) where P(n,x) is the n-th cyclotomic polynomial.

%F a(5*n) = A145467(n). a(5*n + 1) = - A035969(n). a(5*n + 2) = - A145468(n).

%F Convolution inverse of A035959.

%F a(n) = -(1/n)*Sum_{k=1..n} A116073(k)*a(n-k), a(0) = 1. - _Seiichi Manyama_, Mar 25 2017

%e G.f. = 1 - x - x^2 + 2*x^5 - x^6 + 3*x^10 - 2*x^11 - 2*x^12 + 4*x^15 + ...

%e G.f. = 1/q - q^5 - q^11 + 2*q^29 - q^35 + 3*q^59 - 2*q^65 - 2*q^71 + 4*q^89 + ...

%t a[ n_] := SeriesCoefficient[ QPochhammer[ x] / QPochhammer[ x^5], {x, 0, n}]; (* _Michael Somos_, Jun 26 2014 *)

%o (PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A) / eta(x^5 + A), n))};

%Y Cf. A035969, A058511, A145467, A145468.

%K sign

%O 0,6

%A _Michael Somos_, Oct 11 2008