A207834 G.f.: exp( Sum_{n>=1} 5*L(n)*x^n/n ), where L(n) = Fibonacci(n-1)^n + Fibonacci(n+1)^n.

1, 5, 25, 130, 1295, 38861, 4227075, 1309117220, 1123176929475, 2564594183278115, 15604715134340991949, 251021373648740285348860, 10668788238489683954523431475, 1195322752666989652479885363067075, 352750492054485236937115646128341734205

Offset: 0

Author

Paul D. Hanna, Feb 20 2012

Keywords

• nonn

Comments

Given g.f. A(x), note that A(x)^(1/5) is not an integer series.

Compare the definition to the g.f. of the Fibonacci numbers:

1/(1-x-x^2) = exp( Sum_{n>=1} Lucas(n)*x^n/n ), where Lucas(n) = Fibonacci(n-1) + Fibonacci(n+1).

Examples

			G.f.: A(x) = 1 + 5*x + 25*x^2 + 130*x^3 + 1295*x^4 + 38861*x^5 +...
such that, by definition,
log(A(x))/5 = x + 5*x^2/2 + 28*x^3/3 + 641*x^4/4 + 33011*x^5/5 +...+ (Fibonacci(n-1)^n + Fibonacci(n+1)^n)*x^n/n +...
		

Crossrefs

Cf. A207835, A156216, A166168.

Programs

PARI

{L(n)=fibonacci(n-1)^n+fibonacci(n+1)^n}
{a(n)=polcoeff(exp(sum(m=1,n,5*L(m)*x^m/m)+x*O(x^n)),n)}
for(n=0,51,print1(a(n),", "))