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

0,3

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.

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

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}]

Crossrefs

Cf. A000045, A263575.

Sequence in context: A213090 A218225 A279572 * A370196 A370213 A231444

Adjacent sequences: A263573 A263574 A263575 * A263577 A263578 A263579

Keyword

nonn

Author

Vladimir Reshetnikov, Oct 21 2015

Status

approved