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%I #9 Nov 09 2014 07:18:04
%S 1,1,2,5,15,52,207,923,4532,24271,140581,874434,5806557,40955973,
%T 305544958,2402139329,19837601155,171598571288,1550865447043,
%U 14611295961047,143210242799872,1457573997373131,15379106145570681,167962044452359398,1896100883094424657,22096376018936592193
%N G.f.: Sum_{n>=0} x^n / (1 - n*x - n^2*x^2).
%C From _Vaclav Kotesovec_, Nov 09 2014: (Start)
%C (a(n))^(1/n) ~ phi^(1-1/w) * n^(1-1/w) / w^(1-1/w), where w = LambertW(phi*e*n).
%C Limit n->infinity a(n)^(1/n) * LambertW(phi*e*n) / n = phi/e, where phi = (1+sqrt(5))/2 = A001622.
%C (End)
%F a(n) = Sum_{k=0..n} Fibonacci(n-k+1) * k^(n-k).
%e G.f.: A(x) = 1 + x + 2*x^2 + 5*x^3 + 15*x^4 + 52*x^5 + 207*x^6 +...
%e where
%e A(x) = 1 + x/(1-x-x^2) + x^2/(1-2*x-4*x^2) + x^3/(1-3*x-9*x^2) + x^4/(1-4*x-16*x^2) + x^5/(1-5*x-25*x^2) + x^6/(1-6*x-36*x^2) + x^7/(1-7*x-49*x^2) +...
%t Flatten[{1,Table[Sum[Fibonacci[n-k+1] * k^(n-k),{k,0,n}],{n,1,20}]}] (* _Vaclav Kotesovec_, Nov 09 2014 *)
%o (PARI) {a(n)=polcoeff((sum(m=0,n,x^m/(1-m*x-m^2*x^2 +x*O(x^n)))),n)}
%o for(n=0,30,print1(a(n),", "))
%o (PARI) {a(n)=sum(k=0,n,fibonacci(n-k+1)*k^(n-k))}
%o for(n=0,30,print1(a(n),", "))
%Y Cf. A000045.
%K nonn
%O 0,3
%A _Paul D. Hanna_, Nov 08 2014
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