A007552 -id:A007552 - cp's OEIS frontend

A256180 Exponential transform of the Fibonacci numbers.

1, 1, 2, 6, 21, 86, 404, 2121, 12264, 77272, 525941, 3839706, 29891370, 246906569, 2154904856, 19799299506, 190904273049, 1926229186162, 20288311652672, 222568337565537, 2537998989244956, 30029233006187756, 368050599579654557, 4665833729558724030

Offset: 0

Alois P. Heinz, Mar 18 2015

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..500

Cf. A000045, A006701, A007552, A112005.

Row sums of A346415.

F:= n-> (<<0|1>, <1|1>>^n)[1, 2]:
a:= proc(n) option remember; `if`(n=0, 1,
      add(binomial(n-1, j-1) *F(j) *a(n-j), j=1..n))
    end:
seq(a(n), n=0..30);

Table[Sum[BellY[n, k, Fibonacci[Range[n]]], {k, 0, n}], {n, 0, 20}] (* Vladimir Reshetnikov, Nov 09 2016 *)

E.g.f: exp(1/sqrt(5)*(exp((1+sqrt(5))*x/2)-exp((1-sqrt(5))*x/2))).

E.g.f: exp(2/5*sqrt(5)*exp(x/2)*sinh(sqrt(5)*x/2)).

A006701 Exponentiation of g.f. for Fibonacci numbers.

Original entry on oeis.org

0, 1, 1, 5, 13, 60, 246, 1266, 6679, 39568, 247940, 1677435, 12020295, 91463410, 733490265, 6189608760, 54746987035, 506444804075, 4887127598817, 49096724251235, 512474550910080, 5548429401985372, 62208756548406172, 721256031012180537, 8635815672831322186

Offset: 0

N. J. A. Sloane

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..500

Cf. A000045, A007552, A256180.

a[-1] = 1; a[n_] := a[n] = Sum[Binomial[n, k]*Fibonacci[k]*a[n - k - 1], {k, 0, n}]; Table[a[n], {n, 0, 30}] (* Vaclav Kotesovec, Jun 08 2021 *)

a(n) = if (n==-1, 1, sum(k=0, n, binomial(n,k)*fibonacci(k)*a(n-k-1))); \\ Michel Marcus, Jun 11 2017

a(-1) = 1, a(n) = Sum_{k=0..n} binomial(n, k) * A000045(k) * a(n-k-1). - Sean A. Irvine, Jun 11 2017

cp's OEIS Frontend

A256180 Exponential transform of the Fibonacci numbers.

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