A038154 a(n) = n! * Sum_{k=0..n-2} 1/k!.

0, 0, 2, 12, 60, 320, 1950, 13692, 109592, 986400, 9864090, 108505100, 1302061332, 16926797472, 236975164790, 3554627472060, 56874039553200, 966858672404672, 17403456103284402, 330665665962403980, 6613313319248079980, 138879579704209680000

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn

Comments

The number of rank-orderings of (>=2)-element subsets of an n-set. (Counts nontrivial votes in a rank-ordering voting system.) E.g., a(5) = 320 = 120+120+60+20 because of 5-, 4-, 3- and 2-element subsets. - Warren D. Smith, Jul 06 2005

a(n) is the number of simple cycles through a vertex of the complete graph K_(n+1) on n+1 vertices [Hassani]. For example, in the complete graph K_4 with vertex set {A,B,C,D} there are a(3) = 12 simple cycles at the vertex A, namely the six 3-cycles ABCA, ABDA, ACBA, ACDA, ADBA and ADCA and the six 4-cycles ABCDA, ABDCA, ACBDA, ACDBA, ADBCA and ADCBA. The sum of the lengths of the cycles at a vertex of K_n is equal to A141834(n). - Peter Bala, Jul 09 2008

See A000522 for the number of paths between a pair of distinct vertices of K_n. - Peter Bala, Jul 09 2008

a(n) = n*a(n-1) + A000217(n-1), where A000217(n) is the n-th triangular number. - Gary Detlefs, May 20 2010

Examples

			0=1*0+0, 2=2*0+2, 12=3*2+6, 60=4*12+12, 320 = 5*60+20, ... - _Gary Detlefs_, May 20 2010
		

Links

• Mehdi Hassani, Counting and computing by e, arXiv:math/0606613 [math.CO], 2006.
• J. Sawada and A. Williams, Successor rules for flipping pancakes and burnt pancakes, Preprint 2015.
• Index entries for sequences related to factorial numbers

Crossrefs

Cf. A000522, A038155, A141834.

Programs

Mathematica

Table[n!Sum[1/k!,{k,0,n-2}],{n,0,30}] (* Harvey P. Dale, Jun 04 2012 *)

PARI

main(size)=my(k); vector(size,n,(n-1)!*sum(k=0,n-3,1/k!)); \\ Anders Hellström, Jul 14 2015

Formula

a(n) = A007526(n) - n.

a(n) = floor(n!*exp(1))-n-1, n>0. - Vladeta Jovovic, Aug 25 2001

a(n) = n*a(n-1) + n*(n-1), for n>=3, a(2)=2 and a(3) = 12. - Ian Myers, Jul 19 2012

a(n) = A000522(n-2) * n*(n-1). - Doug Bell, Jun 30 2015

E.g.f.: exp(x)*x^2/(1 - x). - Ilya Gutkovskiy, Jan 26 2017

a(n) = 2*A038155(n). - Alois P. Heinz, Jan 26 2017