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A217388 Alternating sums of the ordered Bell numbers (number of preferential arrangements) A000670. 5
1, 0, 3, 10, 65, 476, 4207, 43086, 502749, 6584512, 95663051, 1526969522, 26564598073, 500293750308, 10141049220135, 220142141757718, 5095512540223637, 125275254488912264, 3260259408767933059, 89541327910560478074, 2588146468333823725041 (list; graph; refs; listen; history; text; internal format)
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-Fran├žois 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

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Last modified October 23 21:27 EDT 2021. Contains 348217 sequences. (Running on oeis4.)