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

0,2

Links

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

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

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 *)

Prog

(Maxima)
t(n):=sum(stirling2(n, k)*k!, k, 0, n);
makelist(sum(t(k), k, 0, n), n, 0, 40);
(Magma)
A000670:=func<n | &+[StirlingSecond(n, i)*Factorial(i): i in [0..n]]>;
[&+[A000670(k): k in [0..n]]: n in [0..19]]; // Bruno Berselli, Oct 03 2012
(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

Crossrefs

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

See A239914 for another version.

Sequence in context: A320154 A032273 A143522 * A123310 A058119 A335547

Adjacent sequences: A217386 A217387 A217388 * A217390 A217391 A217392

Keyword

nonn

Author

Emanuele Munarini, Oct 02 2012

Status

approved