A135741 - OEIS

%I #12 Nov 05 2016 12:51:13

%S 1,1,3,7,19,71,347,2115,16395,164071,2099991,34138963,706636219,

%T 18658538939,627640554659,26870678088831,1464622985216331,

%U 101659649883242575,8983370274289495947,1010460953826963080371

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

%H G. C. Greubel, <a href="/A135741/b135741.txt">Table of n, a(n) for n = 0..125</a>

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

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

%F 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

%e 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! +...

%e where

%e 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! +...

%t Flatten[{1,Table[Sum[Binomial[n,k]*Fibonacci[k]^(n-k),{k,0,n}],{n,1,20}]}] (* _Vaclav Kotesovec_, Jul 14 2014 *)

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

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

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

%Y Cf. A135961, A219781.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Nov 27 2007