A145466 Expansion of q^(1/6) * eta(q) / eta(q^5) in powers of q.

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

Offset

0,6

Links

Seiichi Manyama, Table of n, a(n) for n = 0..10000

Formula

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.

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

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.

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.

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).

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

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

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

Convolution inverse of A035959.

a(n) = -(1/n)*Sum_{k=1..n} A116073(k)*a(n-k), a(0) = 1. - Seiichi Manyama, Mar 25 2017

Example

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

Mathematica

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

Prog

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

Crossrefs

Cf. A035969, A058511, A145467, A145468.

Sequence in context: A037855 A037873 A036869 * A036868 A326453 A130116

Adjacent sequences: A145463 A145464 A145465 * A145467 A145468 A145469

Keyword

sign

Author

Michael Somos, Oct 11 2008

Status

approved