cp's OEIS Frontend

A217388 Alternating sums of the ordered Bell numbers (number of preferential arrangements) A000670.

%I A217388 #37 Aug 08 2025 09:09:19
%S A217388 1,0,3,10,65,476,4207,43086,502749,6584512,95663051,1526969522,
%T A217388 26564598073,500293750308,10141049220135,220142141757718,
%U A217388 5095512540223637,125275254488912264,3260259408767933059,89541327910560478074,2588146468333823725041
%N A217388 Alternating sums of the ordered Bell numbers (number of preferential arrangements) A000670.
%H A217388 Vincenzo Librandi, <a href="/A217388/b217388.txt">Table of n, a(n) for n = 0..200</a>
%H A217388 Mathematics Stack Exchange, <a href="https://math.stackexchange.com/a/5088679/929945">Simple closed form for signed partial sums of Fubini numbers</a>, Aug 8 2025.
%F A217388 a(n) = sum((-1)^(n-k)*t(k), k=0..n), where t = A000670 (ordered Bell numbers).
%F A217388 E.g.f.: 1/(2-exp(x))-exp(-x)*log(1/(2-exp(x))). [Typo corrected by _Vaclav Kotesovec_, Oct 08 2013]
%F A217388 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 A217388 a(n) ~ n! /(2*(log(2))^(n+1)). - _Vaclav Kotesovec_, Oct 08 2013
%F A217388 a(n) = Sum_{k=0..n} k!*Stirling2(n+2,k+2)*(2^(k+1)-1)*(-1)^(n-k). - _Mikhail Kurkov_, Aug 08 2025
%p A217388 with(combinat):
%p A217388 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 A217388 t[n_] := Sum[StirlingS2[n, k]k!, {k, 0, n}]; Table[Sum[(-1)^(n - k)t[k], {k, 0, n}], {n, 0, 100}]
%t A217388 (* second program: *)
%t A217388 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 A217388 (Maxima)
%o A217388 t(n):=sum(stirling2(n,k)*k!,k,0,n);
%o A217388 makelist(sum((-1)^(n-k)*t(k),k,0,n),n,0,40);
%o A217388 (Magma)
%o A217388 A000670:=func<n | &+[StirlingSecond(n,i)*Factorial(i): i in [0..n]]>;
%o A217388 [&+[(-1)^(n-k)*A000670(k): k in [0..n]]: n in [0..20]]; // _Bruno Berselli_, Oct 03 2012
%o A217388 (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 A217388 (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
%o A217388 (PARI) a(n) = sum(k=0, n, k!*stirling(n+2,k+2,2)*(2^(k+1)-1)*(-1)^(n-k)) \\ _Mikhail Kurkov_, Aug 08 2025
%Y A217388 Cf. A000670, A006957, A005649, A217389, A217391, A217392.
%K A217388 nonn
%O A217388 0,3
%A A217388 _Emanuele Munarini_, Oct 02 2012