OFFSET
0,3
COMMENTS
From Vaclav Kotesovec, Nov 09 2014: (Start)
(a(n))^(1/n) ~ phi^(1-1/w) * n^(1-1/w) / w^(1-1/w), where w = LambertW(phi*e*n).
Limit n->infinity a(n)^(1/n) * LambertW(phi*e*n) / n = phi/e, where phi = (1+sqrt(5))/2 = A001622.
(End)
LINKS
Robert Israel, Table of n, a(n) for n = 0..550
FORMULA
a(n) = Sum_{k=0..n} Fibonacci(n-k+1) * k^(n-k).
EXAMPLE
G.f.: A(x) = 1 + x + 2*x^2 + 5*x^3 + 15*x^4 + 52*x^5 + 207*x^6 +...
where
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) +...
MAPLE
G:= add(x^n/(1-n*x-n^2*x^2), n=0..40):
S:= series(G, x, 41):
seq(coeff(S, x, i), i=0..40); # Robert Israel, Oct 07 2024
MATHEMATICA
Flatten[{1, Table[Sum[Fibonacci[n-k+1] * k^(n-k), {k, 0, n}], {n, 1, 20}]}] (* Vaclav Kotesovec, Nov 09 2014 *)
PROG
(PARI) {a(n)=polcoeff((sum(m=0, n, x^m/(1-m*x-m^2*x^2 +x*O(x^n)))), n)}
for(n=0, 30, print1(a(n), ", "))
(PARI) {a(n)=sum(k=0, n, fibonacci(n-k+1)*k^(n-k))}
for(n=0, 30, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Nov 08 2014
STATUS
approved