A098799 a(n) = Sum_{k>=0} k^n*A000045(k)/2^(k+1).

1, 5, 47, 665, 12551, 296105, 8382887, 276877865, 10451408231, 443827193705, 20941630652327, 1086925476081065, 61542849621198311, 3775005086748195305, 249368828007507619367, 17649402626956126900265

Offset

0,2

Links

Amiram Eldar, Table of n, a(n) for n = 0..357

Arthur T. Benjamin, Judson D. Neer, Daniel T. Otero and James A. Sellers, A probabilistic view of certain weighted Fibonacci sums, Fibonacci Quarterly, Vol. 41, No. 4 (2002), pp. 360-364.

Spiros D. Dafnis and Andreas N. Philippou, Infinite Sums of Weighted Fibonacci Numbers of Order k, Fibonacci Quart. 54 (2016), no. 2, 149-153.

Formula

G.f.: and closed forms are given in link.

From Amiram Eldar, Jun 16 2020: (Start)

E.g.f.: exp(x)/(4 - 2*exp(x) - exp(2*x)).

a(0) = 1, a(n) = 1 + Sum_{k=0..n-1} binomial(n,k) * (2 + 2^(n-k)) * a(k).

a(n) ~ ((sqrt(5) - 1)/(10 - 2*sqrt(5))) * (1 / log(sqrt(5) - 1))^(n+1) * n!.

a(n) = A103436(n)/2. (End)

Mathematica

a[0] = 1; a[n_] := a[n] = 1 + Sum[Binomial[n, k] * (2^(n - k) + 2) * a[k], {k, 0, n - 1}]; Array[a, 16, 0] (* Amiram Eldar, Jun 16 2020 *)

Crossrefs

Cf. A000045, A103436.

Sequence in context: A127696 A088691 A052802 * A270529 A089155 A254530

Adjacent sequences: A098796 A098797 A098798 * A098800 A098801 A098802

Keyword

nonn

Author

Benoit Cloitre, Oct 05 2004

Extensions

Data corrected by Amiram Eldar, Jun 16 2020

Status

approved