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A263576 Stirling transform of Fibonacci numbers (A000045).

%I A263576 #23 Feb 16 2025 08:33:27
%S A263576 0,1,2,6,23,101,490,2597,14926,92335,610503,4288517,31848677,
%T A263576 249044068,2043448968,17540957166,157108128963,1464813176354,
%U A263576 14187155168782,142469605397465,1480903718595721,15908940627242898,176382950500197589,2015650339677868116
%N A263576 Stirling transform of Fibonacci numbers (A000045).
%H A263576 Alois P. Heinz, <a href="/A263576/b263576.txt">Table of n, a(n) for n = 0..564</a>
%H A263576 Eric Weisstein's MathWorld, <a href="https://mathworld.wolfram.com/FibonacciNumber.html">Fibonacci Number</a>.
%H A263576 Eric Weisstein's MathWorld, <a href="https://mathworld.wolfram.com/StirlingTransform.html">Stirling Transform</a>.
%H A263576 Eric Weisstein's MathWorld, <a href="https://mathworld.wolfram.com/BellPolynomial.html">Bell Polynomial</a>.
%F A263576 a(n) = Sum_{k=0..n} A000045(k)*Stirling2(n,k).
%F A263576 Sum_{k=0..n} a(k)*Stirling1(n,k) = A000045(n).
%F A263576 Let phi=(1+sqrt(5))/2.
%F A263576 a(n) = (B_n(phi)-B_n(1-phi))/sqrt(5), where B_n(x) is n-th Bell polynomial.
%F A263576 2*B_n(phi) = A263575(n) + a(n)*sqrt(5).
%F A263576 E.g.f.: (exp((exp(x)-1)*phi)-exp((exp(x)-1)*(1-phi)))/sqrt(5).
%F A263576 G.f.: Sum_{j>=1} Fibonacci(j)*x^j / Product_{k=1..j} (1 - k*x). - _Ilya Gutkovskiy_, Apr 06 2019
%p A263576 b:= proc(n, m) option remember; `if`(n=0, (<<0|1>,
%p A263576        <1|1>>^m)[1, 2], m*b(n-1, m)+b(n-1, m+1))
%p A263576     end:
%p A263576 a:= n-> b(n, 0):
%p A263576 seq(a(n), n=0..23);  # _Alois P. Heinz_, Aug 03 2021
%t A263576 Table[Sum[Fibonacci[k] StirlingS2[n, k], {k, 0, n}], {n, 0, 23}]
%t A263576 Table[Simplify[(BellB[n, GoldenRatio] - BellB[n, 1 - GoldenRatio])/Sqrt[5]], {n, 0, 23}]
%Y A263576 Cf. A000045, A263575.
%K A263576 nonn
%O A263576 0,3
%A A263576 _Vladimir Reshetnikov_, Oct 21 2015