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Partial sums of the ordered Bell numbers (number of preferential arrangements) A000670.
8

%I #36 Sep 08 2022 08:46:04

%S 1,2,5,18,93,634,5317,52610,598445,7685706,109933269,1732565842,

%T 29824133437,556682481818,11198025452261,241481216430114,

%U 5557135898411469,135927902927547370,3521462566184392693,96323049885512803826

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

%H Vincenzo Librandi, <a href="/A217389/b217389.txt">Table of n, a(n) for n = 0..200</a>

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

%F G.f. = A(x)/(1-x), where A(x) = g.f. for A000670 (see that entry). - _N. J. A. Sloane_, Apr 12 2014

%F a(n) ~ n! / (2* (log(2))^(n+1)). - _Vaclav Kotesovec_, Nov 08 2014

%t t[n_] := Sum[StirlingS2[n, k]k!, {k, 0, n}]; Table[Sum[t[k], {k, 0, n}], {n, 0, 100}]

%t (* second program: *)

%t 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 (Maxima)

%o t(n):=sum(stirling2(n,k)*k!,k,0,n);

%o makelist(sum(t(k),k,0,n),n,0,40);

%o (Magma)

%o A000670:=func<n | &+[StirlingSecond(n,i)*Factorial(i): i in [0..n]]>;

%o [&+[A000670(k): k in [0..n]]: n in [0..19]]; // _Bruno Berselli_, Oct 03 2012

%o (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 Cf. A000670, A006957, A005649, A217388, A217391, A217392.

%Y See A239914 for another version.

%K nonn

%O 0,2

%A _Emanuele Munarini_, Oct 02 2012