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

0,3

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..200

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

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

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 *)

Prog

(Maxima)
t(n):=sum(stirling2(n, k)*k!, k, 0, n);
makelist(sum((-1)^(n-k)*t(k), k, 0, n), n, 0, 40);
(Magma)
A000670:=func<n | &+[StirlingSecond(n, i)*Factorial(i): i in [0..n]]>;
[&+[(-1)^(n-k)*A000670(k): k in [0..n]]: n in [0..20]]; // Bruno Berselli, Oct 03 2012
(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
(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

Crossrefs

Cf. A000670, A006957, A005649, A217389, A217391, A217392.

Sequence in context: A206724 A306187 A009400 * A004102 A072638 A262843

Adjacent sequences: A217385 A217386 A217387 * A217389 A217390 A217391

Keyword

nonn

Author

Emanuele Munarini, Oct 02 2012

Status

approved