A263576 Stirling transform of Fibonacci numbers (A000045).

0, 1, 2, 6, 23, 101, 490, 2597, 14926, 92335, 610503, 4288517, 31848677, 249044068, 2043448968, 17540957166, 157108128963, 1464813176354, 14187155168782, 142469605397465, 1480903718595721, 15908940627242898, 176382950500197589, 2015650339677868116

Offset: 0

Author

Vladimir Reshetnikov, Oct 21 2015

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..564
• Eric Weisstein's MathWorld, Fibonacci Number.
• Eric Weisstein's MathWorld, Stirling Transform.
• Eric Weisstein's MathWorld, Bell Polynomial.

Crossrefs

Cf. A000045, A263575.

Programs

Maple

b:= proc(n, m) option remember; `if`(n=0, (<<0|1>,
       <1|1>>^m)[1, 2], m*b(n-1, m)+b(n-1, m+1))
    end:
a:= n-> b(n, 0):
seq(a(n), n=0..23);  # Alois P. Heinz, Aug 03 2021

Mathematica

Table[Sum[Fibonacci[k] StirlingS2[n, k], {k, 0, n}], {n, 0, 23}]
Table[Simplify[(BellB[n, GoldenRatio] - BellB[n, 1 - GoldenRatio])/Sqrt[5]], {n, 0, 23}]

Formula

a(n) = Sum_{k=0..n} A000045(k)*Stirling2(n,k).

Sum_{k=0..n} a(k)*Stirling1(n,k) = A000045(n).

Let phi=(1+sqrt(5))/2.

a(n) = (B_n(phi)-B_n(1-phi))/sqrt(5), where B_n(x) is n-th Bell polynomial.

2*B_n(phi) = A263575(n) + a(n)*sqrt(5).

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

G.f.: Sum_{j>=1} Fibonacci(j)*x^j / Product_{k=1..j} (1 - k*x). - Ilya Gutkovskiy, Apr 06 2019