A203805 G.f.: exp( Sum_{n>=1} A000204(n)^5 * x^n/n ) where A000204 is the Lucas numbers.

1, 1, 122, 463, 11985, 85456, 1262166, 12018742, 145326748, 1540766090, 17495016342, 191731126832, 2138972609189, 23652975370501, 262682339212290, 2911255335387883, 32296421465575573, 358120616523262016, 3971885483375619384, 44047530724737577400

Offset

0,3

Comments

More generally, exp(Sum_{k>=1} A000204(k)^(2*n+1) * x^k/k) = Product_{k=0..n} 1/(1 - (-1)^(n-k)*A000204(2*k+1)*x - x^2)^binomial(2*n+1,n-k).

Links

G. C. Greubel, Table of n, a(n) for n = 0..950

Formula

G.f.: 1/( (1-x-x^2)^10 * (1+4*x-x^2)^5 * (1-11*x-x^2) ).

G.f.: 1/Product_{n>=1} (1 - Lucas(n)*x^n + (-1)^n*x^(2*n))^A203855(n) where A203855(n) = (1/n)*Sum_{d|n} moebius(n/d)*Lucas(d)^4.

Example

G.f.: A(x) = 1 + x + 122*x^2 + 463*x^3 + 11985*x^4 + 85456*x^5 + ...
where
log(A(x)) = x + 3^5*x^2/2 + 4^5*x^3/3 + 7^5*x^4/4 + 11^5*x^5/5 + 18^5*x^6/6 + 29^5*x^7/7 + 47^5*x^8/8 + ... + Lucas(n)^5*x^n/n + ...

Mathematica

CoefficientList[Series[1/((1 - x - x^2)^10*(1 + 4*x - x^2)^5*(1 - 11*x - x^2)), {x, 0, 50}], x] (* G. C. Greubel, Dec 25 2017 *)

Prog

(PARI) /* Subroutine used in PARI programs below: */
{Lucas(n)=fibonacci(n-1)+fibonacci(n+1)}
(PARI) {a(n)=polcoeff(exp(sum(k=1, n, Lucas(k)^5*x^k/k)+x*O(x^n)), n)}
(PARI) {a(n, m=2)=polcoeff(prod(k=0, m, 1/(1 - (-1)^(m-k)*Lucas(2*k+1)*x - x^2+x*O(x^n))^binomial(2*m+1, m-k)), n)}

Crossrefs

Cf. A002571, A203803, A203804, A203806, A203807, A203808, A203809.

Cf. A203855, A203800.

Sequence in context: A070955 A195856 A116216 * A275270 A031700 A173267

Adjacent sequences: A203802 A203803 A203804 * A203806 A203807 A203808

Keyword

nonn

Author

Paul D. Hanna, Jan 06 2012

Status

approved