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A263576 Stirling transform of Fibonacci numbers (A000045). 5
0, 1, 2, 6, 23, 101, 490, 2597, 14926, 92335, 610503, 4288517, 31848677, 249044068, 2043448968, 17540957166, 157108128963, 1464813176354, 14187155168782, 142469605397465, 1480903718595721, 15908940627242898, 176382950500197589, 2015650339677868116 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

LINKS

Table of n, a(n) for n=0..23.

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

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 * A231444 A248900 A120346

Adjacent sequences:  A263573 A263574 A263575 * A263577 A263578 A263579

KEYWORD

nonn

AUTHOR

Vladimir Reshetnikov, Oct 21 2015

STATUS

approved

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Last modified July 24 05:07 EDT 2019. Contains 325290 sequences. (Running on oeis4.)