A217388 Alternating sums of the ordered Bell numbers (number of preferential arrangements) A000670.
1, 0, 3, 10, 65, 476, 4207, 43086, 502749, 6584512, 95663051, 1526969522, 26564598073, 500293750308, 10141049220135, 220142141757718, 5095512540223637, 125275254488912264, 3260259408767933059, 89541327910560478074, 2588146468333823725041
Offset: 0
Keywords
Links
- Vincenzo Librandi, Table of n, a(n) for n = 0..200
- Mathematics Stack Exchange, Simple closed form for signed partial sums of Fubini numbers, Aug 8 2025.
Programs
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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
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Magma
A000670:=func
; [&+[(-1)^(n-k)*A000670(k): k in [0..n]]: n in [0..20]]; // Bruno Berselli, Oct 03 2012 -
Maple
with(combinat): 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
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Mathematica
t[n_] := Sum[StirlingS2[n, k]k!, {k, 0, n}]; Table[Sum[(-1)^(n - k)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; a[n_] := Sum[(-1)^(n-k) Fubini[k, 1], {k, 0, n}]; Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Mar 31 2016 *) -
Maxima
t(n):=sum(stirling2(n,k)*k!,k,0,n); makelist(sum((-1)^(n-k)*t(k),k,0,n),n,0,40);
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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
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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
Formula
a(n) = sum((-1)^(n-k)*t(k), k=0..n), where t = A000670 (ordered Bell numbers).
E.g.f.: 1/(2-exp(x))-exp(-x)*log(1/(2-exp(x))). [Typo corrected by Vaclav Kotesovec, Oct 08 2013]
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
a(n) ~ n! /(2*(log(2))^(n+1)). - Vaclav Kotesovec, Oct 08 2013
a(n) = Sum_{k=0..n} k!*Stirling2(n+2,k+2)*(2^(k+1)-1)*(-1)^(n-k). - Mikhail Kurkov, Aug 08 2025