A249892 - OEIS

A249892

G.f.: Sum_{n>=0} x^n / (1 - n*x - n^2*x^2).

1, 1, 2, 5, 15, 52, 207, 923, 4532, 24271, 140581, 874434, 5806557, 40955973, 305544958, 2402139329, 19837601155, 171598571288, 1550865447043, 14611295961047, 143210242799872, 1457573997373131, 15379106145570681, 167962044452359398, 1896100883094424657, 22096376018936592193

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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