A082491 a(n) = n! * d(n), where n! = factorial numbers (A000142), d(n) = subfactorial numbers (A000166).

1, 0, 2, 12, 216, 5280, 190800, 9344160, 598066560, 48443028480, 4844306476800, 586161043776000, 84407190782745600, 14264815236056985600, 2795903786354347468800, 629078351928420506112000, 161044058093696572354560000, 46541732789077953723039744000

Offset

0,3

Comments

a(n) is also the number of pairs of n-permutations p and q such that p(x)<>q(x) for each x in { 1, 2, ..., n }.

Or number of n X n matrices with exactly one 1 and one 2 in each row and column, other entries 0 (cf. A001499). - Vladimir Shevelev, Mar 22 2010

a(n) is approximately equal to (n!)^2/e. - J. M. Bergot, Jun 09 2018

Links

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

Ira Gessel, Enumerative applications of symmetric functions, Séminaire Lotharingien de Combinatoire, B17a (1987), 17 pp.

Shawn L. Witte, Link Nomenclature, Random Grid Diagrams, and Markov Chain Methods in Knot Theory, Ph. D. Dissertation, University of California-Davis (2020).

Formula

a(n) = n! * d(n) where d(n) = A000166(n).

a(n) = Sum_{k=0..n} binomial(n, k)^2 * (-1)^k * (n - k)!^2 * k!.

a(n+2) = (n+2)*(n+1) * ( a(n+1) + (n+1)*a(n) ).

a(n) ~ 2*Pi*n^(2*n+1)*exp(-2*n-1). - Ilya Gutkovskiy, Dec 04 2016

Maple

with (combstruct):a:=proc(m) [ZL, {ZL=Set(Cycle(Z, card>=m))}, labeled]; end: ZLL:=a(2):seq(count(ZLL, size=n)*n!, n=0..15); # Zerinvary Lajos, Jun 11 2008

Mathematica

Table[Subfactorial[n]*n!, {n, 0, 15}] (* Zerinvary Lajos, Jul 10 2009 *)

Prog

(Maxima) A000166[0]:1$
A000166[n]:=n*A000166[n-1]+(-1)^n$
  makelist(n!*A000166[n], n, 0, 12); /* Emanuele Munarini, Mar 01 2011 */
(PARI)
d(n)=if(n<1, n==0, n*d(n-1)+(-1)^n);
a(n)=d(n)*n!;
vector(33, n, a(n-1))
/* Joerg Arndt, May 28 2012 */
(PARI) {a(n) = if( n<2, n==0, n! * round(n! / exp(1)))}; /* Michael Somos, Jun 24 2018 */
(Python)
A082491_list, m, x = [], 1, 1
for n in range(10*2):
....x, m = x*n**2 + m, -(n+1)*m
....A082491_list.append(x) # Chai Wah Wu, Nov 03 2014
(Scala)
val A082491_pairs: LazyList[BigInt && BigInt] =
  (BigInt(0), BigInt(1)) #::
  (BigInt(1), BigInt(0)) #::
  lift2 {
    case ((n, z), (_, y)) =>
      (n+2, (n+2)*(n+1)*((n+1)*z+y))
  } (A082491_pairs, A082491_pairs.tail)
val A082491: LazyList[BigInt] =
  lift1(_._2)(A082491_pairs)
/** Luc Duponcheel, Jan 25 2020 */

Crossrefs

Cf. A000142, A000166.

Sequence in context: A156489 A129893 A008352 * A292812 A153302 A123118

Adjacent sequences: A082488 A082489 A082490 * A082492 A082493 A082494

Keyword

easy,nonn

Author

Emanuele Munarini, Apr 28 2003

Status

approved