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A232472 2-Fubini numbers. 10
2, 10, 62, 466, 4142, 42610, 498542, 6541426, 95160302, 1520385010, 26468935022, 498766780786, 10114484622062, 219641848007410, 5085371491003502, 125055112347154546, 3255163896227709422, 89416052656071565810, 2584886208925055791982, 78447137202259689678706, 2493719594804686310662382 (list; graph; refs; listen; history; text; internal format)
OFFSET

2,1

LINKS

Table of n, a(n) for n=2..22.

Andrei Z. Broder, The r-Stirling numbers, Discrete Math. 49, 241-259 (1984).

I. Mezo, Periodicity of the last digits of some combinatorial sequences, J. Integer Seq. 17, Article 14.1.1 (2014).

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)

EXAMPLE

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

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-Fran├žois Alcover, Mar 30 2016 *)

PROG

(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]]>;

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

CROSSREFS

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

Column k=1 of A122101.

Sequence in context: A192942 A307364 A141140 * A175962 A183165 A129130

Adjacent sequences:  A232469 A232470 A232471 * A232473 A232474 A232475

KEYWORD

nonn,easy

AUTHOR

N. J. A. Sloane, Nov 27 2013

STATUS

approved

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Last modified July 28 11:20 EDT 2021. Contains 346326 sequences. (Running on oeis4.)