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%I #67 Jan 20 2024 09:02:33
%S 2,10,62,466,4142,42610,498542,6541426,95160302,1520385010,
%T 26468935022,498766780786,10114484622062,219641848007410,
%U 5085371491003502,125055112347154546,3255163896227709422,89416052656071565810,2584886208925055791982,78447137202259689678706,2493719594804686310662382
%N 2-Fubini numbers.
%H S. Alex Bradt, Jennifer Elder, Pamela E. Harris, Gordon Rojas Kirby, Eva Reutercrona, Yuxuan (Susan) Wang, and Juliet Whidden, <a href="https://arxiv.org/abs/2401.06937">Unit interval parking functions and the r-Fubini numbers</a>, arXiv:2401.06937 [math.CO], 2024. See page 8.
%H Andrei Z. Broder, <a href="http://dx.doi.org/10.1016/0012-365X(84)90161-4">The r-Stirling numbers</a>, Discrete Math. 49, 241-259 (1984).
%H Eldar Fischer, Johann A. Makowsky, and Vsevolod Rakita, <a href="https://arxiv.org/abs/2302.08265">MC-finiteness of restricted set partition functions</a>, arXiv:2302.08265 [math.CO], 2023.
%H I. Mezo, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL17/Mezo/mezo19.html">Periodicity of the last digits of some combinatorial sequences</a>, J. Integer Seq. 17, Article 14.1.1 (2014).
%F Let A(x) be the g.f. A232472, B(x) the g.f. A000670, then A(x) = (1-x)*B(x) - 1. - _Sergei N. Gladkovskii_, Nov 29 2013
%F a(n) = Sum_{k>=2} T_k*k^(n-2)/2^k where T_k is the (k-1)-st triangular number (i.e., T_k = k*(k-1)/2). - _Derek Orr_, Jan 01 2016
%F a(n) = 2*A069321(n-1). - _Vincenzo Librandi_, Jan 03 2016, corrected by _Vaclav Kotesovec_, Jul 01 2018
%F a(n) ~ n! / (2 * (log(2))^(n+1)). - _Vaclav Kotesovec_, Jul 01 2018
%F From _Peter Bala_, Dec 08 2020: (Start)
%F a(n+2) = Sum_{k = 0..n} (k+2)!/k!*( Sum{i = 0..k} (-1)^(k-i)*binomial(k,i)*(i+2)^n ).
%F a(n+2) = Sum_{k = 0..n} 2^(n-k)*binomial(n,k)*( Sum_{i = 0..k} Stirling2(k,i)*(i+2)! ).
%F a(n) = 2*A069321(n-1) = A000670(n) - A000670(n-1).
%F a(n+1)= (1/2)*Sum_{k = 0..n} binomial(n,k)*A000670(k+1) for n >= 1.
%F E.g.f. with offset 0: 2*exp(2*z)/(2 - exp(z))^3 = 2 + 10*z + 62*z^2/2! + 466*z^3/3! + .... (End)
%e G.f.: 2*x^2 + 10*x^3 + 62*x^4 + 466*x^5 + 4142*x^6 + 42610*x^7 + 498542*x^8 + ...
%p # r-Stirling numbers of second kind (e.g., A008277, A143494, A143495):
%p T := (n,k,r) -> (1/(k-r)!)*add ((-1)^(k+i+r)*binomial(k-r,i)*(i+r)^(n-r),i = 0..k-r):
%p # r-Bell numbers (e.g. A000110, A005493, A005494):
%p B := (n,r) -> add(T(n,k,r),k=r..n);
%p SB := r -> [seq(B(n,r),n=r..30)];
%p SB(2);
%p # r-Fubini numbers (e.g., A000670, A232472, A232473, A232474):
%p F := (n,r) -> add((k)!*T(n,k,r),k=r..n);
%p SF := r -> [seq(F(n,r),n=r..30)];
%p SF(2);
%t Rest[max=20; t=Sum[n^(n - 1) x^n / n!, {n, 1, max}]; 2 Range[0, max]!CoefficientList[Series[D[1 / (1 - y (Exp[x] - 1)), y] /.y->1, {x, 0, max}], x]] (* _Vincenzo Librandi_ Jan 03 2016 *)
%t Fubini[n_, r_] := Sum[k!*Sum[(-1)^(i+k+r)*(i+r)^(n-r)/(i!*(k-i-r)!), {i, 0, k-r}], {k, r, n}]; Table[Fubini[n, 2], {n, 2, 22}] (* _Jean-François Alcover_, Mar 30 2016 *)
%o (Magma) r:=2; r_Fubini:=func<n,r | &+[Factorial(k)*&+[(-1)^(k+h+r)*(h+r)^(n-r)/(Factorial(h)*Factorial(k-h-r)): h in [0..k-r]]: k in [r..n]]>;
%o [r_Fubini(n, r): n in [r..22]]; // _Bruno Berselli_, Mar 30 2016
%Y Cf. A000110, A000670, A005493, A005494, A008277, A069321, A143494, A143495, A232472, A232473, A232474.
%Y Column k=1 of A122101.
%K nonn,easy
%O 2,1
%A _N. J. A. Sloane_, Nov 27 2013
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