A203809 G.f.: exp( Sum_{n>=1} A000204(n)^9 * x^n/n ) where A000204 is the Lucas numbers.

1, 1, 9842, 97223, 58608265, 1390114224, 296390076414, 12122505505998, 1486321234837932, 84428445979241330, 7833461016478812734, 528228569507280147664, 43275470600883540869733, 3148637876123977595284117, 245565185017744596492591850

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 + 9842*x^2 + 97223*x^3 + 58608265*x^4 + 1390114224*x^5 + ...
where
log(A(x)) = x + 3^9*x^2/2 + 4^9*x^3/3 + 7^9*x^4/4 + 11^9*x^5/5 + 18^9*x^6/6 + 29^9*x^7/7 + 47^9*x^8/8 + ... + Lucas(n)^9*x^n/n + ...
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..530
• Index entries for linear recurrences with constant coefficients, order 512.

Crossrefs

Cf. A002571, A203803, A203804, A203805, A203806, A203807, A203808.

Cf. A203859, A203800.

Programs

Mathematica

CoefficientList[Series[1/((1 - x - x^2)^126*(1 + 4*x - x^2)^84*(1 - 11*x - x^2)^36*(1 + 29*x - x^2)^9*(1 - 76*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)^9*x^k/k)+x*O(x^n)), n)}

PARI

{a(n,m=4)=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)^126 * (1+4*x-x^2)^84 * (1-11*x-x^2)^36 * (1+29*x-x^2)^9 * (1-76*x-x^2) ).

G.f.: 1/Product_{n>=1} (1 - Lucas(n)*x^n + (-1)^n*x^(2*n))^A203859(n) where A203859(n) = (1/n)*Sum_{d|n} moebius(n/d)*Lucas(d)^8.