A203808 G.f.: exp( Sum_{n>=1} A000204(n)^8 * x^n/n ) where A000204 is the Lucas numbers.

1, 1, 3281, 25126, 6845526, 121368902, 12805025677, 373879862237, 24707348223677, 948781359159752, 50702478932197928, 2210812262034197128, 108528095366637700218, 4974402150387759436378, 236926456045384849970778, 11047772769135934828000404

Offset: 0

Author

Paul D. Hanna, Jan 06 2012

Keywords

• nonn

Comments

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

Examples

			G.f.: A(x) = 1 + x + 3281*x^2 + 25126*x^3 + 6845526*x^4 + 121368902*x^5 + ...
where
log(A(x)) = x + 3^8*x^2/2 + 4^8*x^3/3 + 7^8*x^4/4 + 11^8*x^5/5 + 18^8*x^6/6 + 29^8*x^7/7 + 47^8*x^8/8 + ... + Lucas(n)^8*x^n/n + ...
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..596
• Index entries for linear recurrences with constant coefficients, order 256.

Crossrefs

Cf. A002571, A203803, A203804, A203805, A203806, A203807, A203809.

Cf. A203858, A203800.

Programs

Mathematica

CoefficientList[Series[1/((1 - x)^70*(1 + 3*x + x^2)^56*(1 - 7*x + x^2)^28*(1 + 18*x + x^2)^8*(1 - 47*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)^8*x^k/k)+x*O(x^n)), n)}

PARI

{a(n,m=4)=polcoeff(1/(1 - (-1)^m*x+x*O(x^n))^binomial(2*m,m) * prod(k=1,m,1/(1 - (-1)^(m-k)*Lucas(2*k)*x + x^2+x*O(x^n))^binomial(2*m,m-k)),n)}

Formula

G.f.: 1/( (1-x)^70 * (1+3*x+x^2)^56 * (1-7*x+x^2)^28 * (1+18*x+x^2)^8 * (1-47*x+x^2) ).

G.f.: 1/Product_{n>=1} (1 - Lucas(n)*x^n + (-1)^n*x^(2*n))^A203858(n) where A203858(n) = (1/n)*Sum_{d|n} moebius(n/d)*Lucas(d)^7.