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%I #33 Sep 08 2022 08:46:04
%S 1,0,3,10,65,476,4207,43086,502749,6584512,95663051,1526969522,
%T 26564598073,500293750308,10141049220135,220142141757718,
%U 5095512540223637,125275254488912264,3260259408767933059,89541327910560478074,2588146468333823725041
%N Alternating sums of the ordered Bell numbers (number of preferential arrangements) A000670.
%H Vincenzo Librandi, <a href="/A217388/b217388.txt">Table of n, a(n) for n = 0..200</a>
%F a(n) = sum((-1)^(n-k)*t(k), k=0..n), where t = A000670 (ordered Bell numbers).
%F E.g.f.: 1/(2-exp(x))-exp(-x)*log(1/(2-exp(x))). [Typo corrected by _Vaclav Kotesovec_, Oct 08 2013]
%F G.f.: 1/(1+x)/Q(0), where Q(k)= 1 - x*(k+1)/(1 - x*(2*k+2)/Q(k+1)); (continued fraction). - _Sergei N. Gladkovskii_, May 20 2013
%F a(n) ~ n! /(2*(log(2))^(n+1)). - _Vaclav Kotesovec_, Oct 08 2013
%p with(combinat):
%p seq(sum((-1)^(n-k)*sum(factorial(j)*stirling2(k,j), j=0..k), k=0..n), n=0..30); # _Muniru A Asiru_, Feb 07 2018
%t t[n_] := Sum[StirlingS2[n, k]k!, {k, 0, n}]; Table[Sum[(-1)^(n - k)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; a[n_] := Sum[(-1)^(n-k) Fubini[k, 1], {k, 0, n}]; Table[a[n], {n, 0, 20}] (* _Jean-François Alcover_, Mar 31 2016 *)
%o (Maxima)
%o t(n):=sum(stirling2(n,k)*k!,k,0,n);
%o makelist(sum((-1)^(n-k)*t(k),k,0,n),n,0,40);
%o (Magma)
%o A000670:=func<n | &+[StirlingSecond(n,i)*Factorial(i): i in [0..n]]>;
%o [&+[(-1)^(n-k)*A000670(k): k in [0..n]]: n in [0..20]]; // _Bruno Berselli_, Oct 03 2012
%o (PARI) for(n=0,30, print1(sum(k=0,n, (-1)^(n-k)*sum(j=0,k, j!*stirling(k,j,2))), ", ")) \\ _G. C. Greubel_, Feb 07 2018
%o (GAP) List([0..30],n->Sum([0..n],k->(-1)^(n-k)*Sum([0..k], j-> Factorial(j)*Stirling2(k,j)))); # _Muniru A Asiru_, Feb 07 2018
%Y Cf. A000670, A006957, A005649, A217389, A217391, A217392.
%K nonn
%O 0,3
%A _Emanuele Munarini_, Oct 02 2012
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