A232472 2-Fubini numbers.

2, 10, 62, 466, 4142, 42610, 498542, 6541426, 95160302, 1520385010, 26468935022, 498766780786, 10114484622062, 219641848007410, 5085371491003502, 125055112347154546, 3255163896227709422, 89416052656071565810, 2584886208925055791982, 78447137202259689678706, 2493719594804686310662382

Offset: 2

Author

N. J. A. Sloane, Nov 27 2013

Keywords

• nonn
• easy

Examples

			G.f.: 2*x^2 + 10*x^3 + 62*x^4 + 466*x^5 + 4142*x^6 + 42610*x^7 + 498542*x^8 + ...
		

Links

• S. Alex Bradt, Jennifer Elder, Pamela E. Harris, Gordon Rojas Kirby, Eva Reutercrona, Yuxuan (Susan) Wang, and Juliet Whidden, Unit interval parking functions and the r-Fubini numbers, arXiv:2401.06937 [math.CO], 2024. See page 8.
• Andrei Z. Broder, The r-Stirling numbers, Discrete Math. 49, 241-259 (1984).
• Eldar Fischer, Johann A. Makowsky, and Vsevolod Rakita, MC-finiteness of restricted set partition functions, arXiv:2302.08265 [math.CO], 2023.
• I. Mezo, Periodicity of the last digits of some combinatorial sequences, J. Integer Seq. 17, Article 14.1.1 (2014).

Crossrefs

Cf. A000110, A000670, A005493, A005494, A008277, A069321, A143494, A143495, A232472, A232473, A232474.

Column k=1 of A122101.

Programs

Magma

r:=2; r_Fubini:=func;
[r_Fubini(n, r): n in [r..22]]; // Bruno Berselli, Mar 30 2016

Maple

# r-Stirling numbers of second kind (e.g., A008277, A143494, A143495):
T := (n,k,r) -> (1/(k-r)!)*add ((-1)^(k+i+r)*binomial(k-r,i)*(i+r)^(n-r),i = 0..k-r):
# r-Bell numbers (e.g. A000110, A005493, A005494):
B := (n,r) -> add(T(n,k,r),k=r..n);
SB := r -> [seq(B(n,r),n=r..30)];
SB(2);
# r-Fubini numbers (e.g., A000670, A232472, A232473, A232474):
F := (n,r) -> add((k)!*T(n,k,r),k=r..n);
SF := r -> [seq(F(n,r),n=r..30)];
SF(2);

Mathematica

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 *)
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 *)

Formula

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

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

a(n) = 2*A069321(n-1). - Vincenzo Librandi, Jan 03 2016, corrected by Vaclav Kotesovec, Jul 01 2018

a(n) ~ n! / (2 * (log(2))^(n+1)). - Vaclav Kotesovec, Jul 01 2018

From Peter Bala, Dec 08 2020: (Start)

a(n+2) = Sum_{k = 0..n} (k+2)!/k!*( Sum{i = 0..k} (-1)^(k-i)*binomial(k,i)*(i+2)^n ).

a(n+2) = Sum_{k = 0..n} 2^(n-k)*binomial(n,k)*( Sum_{i = 0..k} Stirling2(k,i)*(i+2)! ).

a(n) = 2*A069321(n-1) = A000670(n) - A000670(n-1).

a(n+1)= (1/2)*Sum_{k = 0..n} binomial(n,k)*A000670(k+1) for n >= 1.

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)