A217389 Partial sums of the ordered Bell numbers (number of preferential arrangements) A000670.

1, 2, 5, 18, 93, 634, 5317, 52610, 598445, 7685706, 109933269, 1732565842, 29824133437, 556682481818, 11198025452261, 241481216430114, 5557135898411469, 135927902927547370, 3521462566184392693, 96323049885512803826, 2774010846129897006941, 83898835844633970888762

Offset: 0

Author

Emanuele Munarini, Oct 02 2012

Keywords

• nonn

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..200

Crossrefs

Cf. A000670, A006957, A005649, A217388, A217391, A217392.

See A239914 for another version.

Programs

Magma

A000670:=func;
[&+[A000670(k): k in [0..n]]: n in [0..19]]; // Bruno Berselli, Oct 03 2012

Maple

b:= proc(n, k) option remember;
     `if`(n=0, k!, k*b(n-1, k)+b(n-1, k+1))
    end:
a:= proc(n) option remember; `if`(n<0, 0, a(n-1)+b(n, 0)) end:
seq(a(n), n=0..23);  # Alois P. Heinz, Feb 20 2025

Mathematica

t[n_] := Sum[StirlingS2[n, k]k!, {k, 0, n}]; Table[Sum[t[k], {k, 0, n}], {n, 0, 100}]
(* second program: *)
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}]; Fubini[0, 1] = 1; Table[Fubini[n, 1], {n, 0, 20}] // Accumulate (* Jean-François Alcover, Mar 31 2016 *)

Maxima

t(n):=sum(stirling2(n,k)*k!,k,0,n);
makelist(sum(t(k),k,0,n),n,0,40);

PARI

for(n=0,30, print1(sum(k=0,n, sum(j=0,k, j!*stirling(k,j,2))), ", ")) \\ G. C. Greubel, Feb 07 2018

Formula

a(n) = Sum_{k=0..n} t(k), where t = A000670 (ordered Bell numbers).

G.f. = A(x)/(1-x), where A(x) = g.f. for A000670 (see that entry). - N. J. A. Sloane, Apr 12 2014

a(n) ~ n! / (2* (log(2))^(n+1)). - Vaclav Kotesovec, Nov 08 2014