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

Author

Paul D. Hanna, Jan 06 2012

Keywords

• nonn

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).

Examples

			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 + ...
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..950
• Index entries for linear recurrences with constant coefficients, signature (1,121,220,-3460,-5932,52717,52667,-483925,-81600,2532240,-1172640,-6764090,4911050,11191850,-8809960,-13039640,8809960,11191850,-4911050,-6764090,1172640,2532240,81600,-483925,-52667,52717,5932,-3460,-220,121,-1,-1).

Crossrefs

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

Cf. A203855, A203800.

Programs

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 *)

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)}

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.