cp's OEIS Frontend

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

%I A217389 #39 Feb 20 2025 09:26:35
%S A217389 1,2,5,18,93,634,5317,52610,598445,7685706,109933269,1732565842,
%T A217389 29824133437,556682481818,11198025452261,241481216430114,
%U A217389 5557135898411469,135927902927547370,3521462566184392693,96323049885512803826,2774010846129897006941,83898835844633970888762
%N A217389 Partial sums of the ordered Bell numbers (number of preferential arrangements) A000670.
%H A217389 Vincenzo Librandi, <a href="/A217389/b217389.txt">Table of n, a(n) for n = 0..200</a>
%F A217389 a(n) = Sum_{k=0..n} t(k), where t = A000670 (ordered Bell numbers).
%F A217389 G.f. = A(x)/(1-x), where A(x) = g.f. for A000670 (see that entry). - _N. J. A. Sloane_, Apr 12 2014
%F A217389 a(n) ~ n! / (2* (log(2))^(n+1)). - _Vaclav Kotesovec_, Nov 08 2014
%p A217389 b:= proc(n, k) option remember;
%p A217389      `if`(n=0, k!, k*b(n-1, k)+b(n-1, k+1))
%p A217389     end:
%p A217389 a:= proc(n) option remember; `if`(n<0, 0, a(n-1)+b(n, 0)) end:
%p A217389 seq(a(n), n=0..23);  # _Alois P. Heinz_, Feb 20 2025
%t A217389 t[n_] := Sum[StirlingS2[n, k]k!, {k, 0, n}]; Table[Sum[t[k], {k, 0, n}], {n, 0, 100}]
%t A217389 (* second program: *)
%t A217389 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 *)
%o A217389 (Maxima)
%o A217389 t(n):=sum(stirling2(n,k)*k!,k,0,n);
%o A217389 makelist(sum(t(k),k,0,n),n,0,40);
%o A217389 (Magma)
%o A217389 A000670:=func<n | &+[StirlingSecond(n,i)*Factorial(i): i in [0..n]]>;
%o A217389 [&+[A000670(k): k in [0..n]]: n in [0..19]]; // _Bruno Berselli_, Oct 03 2012
%o A217389 (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
%Y A217389 Cf. A000670, A006957, A005649, A217388, A217391, A217392.
%Y A217389 See A239914 for another version.
%K A217389 nonn
%O A217389 0,2
%A A217389 _Emanuele Munarini_, Oct 02 2012