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A263576 Stirling transform of Fibonacci numbers (A000045). 6
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
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
Sequence in context: A213090 A218225 A279572 * A370196 A370213 A231444
KEYWORD
nonn
AUTHOR
STATUS
approved

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Last modified April 23 06:04 EDT 2024. Contains 371906 sequences. (Running on oeis4.)