A263576 - OEIS

%I #22 Aug 21 2021 21:42:35

%S 0,1,2,6,23,101,490,2597,14926,92335,610503,4288517,31848677,

%T 249044068,2043448968,17540957166,157108128963,1464813176354,

%U 14187155168782,142469605397465,1480903718595721,15908940627242898,176382950500197589,2015650339677868116

%N Stirling transform of Fibonacci numbers (A000045).

%H Alois P. Heinz, <a href="/A263576/b263576.txt">Table of n, a(n) for n = 0..564</a>

%H Eric Weisstein's MathWorld, <a href="http://mathworld.wolfram.com/FibonacciNumber.html">Fibonacci Number</a>.

%H Eric Weisstein's MathWorld, <a href="http://mathworld.wolfram.com/StirlingTransform.html">Stirling Transform</a>.

%H Eric Weisstein's MathWorld, <a href="http://mathworld.wolfram.com/BellPolynomial.html">Bell Polynomial</a>.

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

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

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

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

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

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

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

%p b:= proc(n, m) option remember; `if`(n=0, (<<0|1>,

%p        <1|1>>^m)[1, 2], m*b(n-1, m)+b(n-1, m+1))

%p     end:

%p a:= n-> b(n, 0):

%p seq(a(n), n=0..23);  # _Alois P. Heinz_, Aug 03 2021

%t Table[Sum[Fibonacci[k] StirlingS2[n, k], {k, 0, n}], {n, 0, 23}]

%t Table[Simplify[(BellB[n, GoldenRatio] - BellB[n, 1 - GoldenRatio])/Sqrt[5]], {n, 0, 23}]

%Y Cf. A000045, A263575.

%K nonn

%O 0,3

%A _Vladimir Reshetnikov_, Oct 21 2015