A135741 E.g.f.: A(x) = Sum_{n>=0} exp(Fibonacci(n)*x) * x^n/n!.

1, 1, 3, 7, 19, 71, 347, 2115, 16395, 164071, 2099991, 34138963, 706636219, 18658538939, 627640554659, 26870678088831, 1464622985216331, 101659649883242575, 8983370274289495947, 1010460953826963080371

Offset: 0

Author

Paul D. Hanna, Nov 27 2007

Keywords

• nonn

Examples

			E.g.f.: A(x) = 1 + x + 3*x^2/2! + 7*x^3/3! + 19*x^4/4! + 71*x^5/5! + 347*x^6/6! +...
where
A(x) = 1 + exp(x)*x + exp(x)*x^2/2! + exp(2*x)*x^3/3! + exp(3*x)*x^4/4! + exp(5*x)*x^5/5! + exp(8*x)*x^6/6! + exp(13*x)*x^7/7! + exp(21*x)*x^8/8! +...
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..125

Crossrefs

Cf. A135961, A219781.

Programs

Mathematica

Flatten[{1,Table[Sum[Binomial[n,k]*Fibonacci[k]^(n-k),{k,0,n}],{n,1,20}]}] (* Vaclav Kotesovec, Jul 14 2014 *)

PARI

{a(n)=sum(k=0,n,binomial(n,k)*fibonacci(k)^(n-k))}

PARI

{a(n)=n!*polcoeff(sum(k=0,n,exp(fibonacci(k)*x +x*O(x^n))*x^k/k!),n)}

PARI

{a(n)=polcoeff(sum(k=0, n, x^k/(1-fibonacci(k)*x +x*O(x^n))^(k+1)), n)}

Formula

a(n) = Sum_{k=0..n} C(n,k) * Fibonacci(k)^(n-k).

O.g.f.: Sum_{n>=0} x^n/(1 - Fibonacci(n)*x)^(n+1).

a(n) ~ c * 2^(n+1/2) * ((1+sqrt(5))/2)^(n^2/4) / (sqrt(Pi*n) * 5^(n/4)), where c = Sum_{k=-infinity..infinity} 5^(k/2)*((1+sqrt(5))/2)^(-k^2) = 3.5769727481316948565395...(see A219781) if n is even, and c = Sum_{k=-infinity..infinity} 5^((k+1/2)/2)*((1+sqrt(5))/2)^(-(k+1/2)^2) = 3.5769727390073366345992... if n is odd. - Vaclav Kotesovec, Jul 14 2014