A013999 From applying the "rational mean" to the number e.

1, 1, 2, 8, 42, 258, 1824, 14664, 132360, 1326120, 14606640, 175448160, 2282469840, 31972303440, 479793807360, 7679384173440, 130586660507520, 2351111258805120, 44679858911251200, 893744703503769600, 18771276190401504000, 413017883356110278400

Offset: 0

Author

Domingo Gomez Morin (Dgomezm(AT)etheron.net)

Keywords

• nonn

Comments

Binomial transform of A000271. - Vladeta Jovovic, Jun 26 2007

Conjecture: this is also the number of acyclic orientations of the complement of the path graph. - Martin Rubey, Oct 15 2023

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..200
• Domingo Gómez Morín, New Elements For The Irrational Numbers, Journal of Transfigural Mathematics, Vol. 2, No. 1, 1996.
• R. P. Stanley, An Equivalence Relation on the Symmetric Group and Multiplicity-free Flag h-Vectors, preprint, 2008. - From _N. J. A. Sloane_, May 06 2012

Crossrefs

Cf. A000271.

Programs

Mathematica

Table[SeriesCoefficient[Sum[k!*(x*(1-x))^k,{k,0,n}],{x,0,n}],{n,1,20}] (* Vaclav Kotesovec, Oct 07 2012 *)

Maxima

makelist(sum(binomial(n-k+1,k)*(-1)^k*(n-k+1)!,k,0,floor((n+1)/2)),n,0,20); /* Emanuele Munarini, Jul 01 2013 */

Formula

G.f.: Sum_{n>=0} n!*(x*(1-x))^n. - Vladeta Jovovic, Jun 26 2007

Recurrence: a(n) = (n+3)*a(n-1) - (2*n+1)*a(n-2) + n*a(n-3). - Vaclav Kotesovec, Oct 07 2012

G.f.: 1/Q(0), where Q(k)= 1 + x/(1-x) - x*(k+2)/(1 - x*(k+1)/Q(k+1)); (continued fraction). - Sergei N. Gladkovskii, Apr 21 2013

a(n) = sum(binomial(n-k+1,k)*(-1)^k*(n-k+1)!, k=0..floor((n+1)/2)). - Emanuele Munarini, Jul 01 2013

a(n) ~ n!*n/exp(1). - Vaclav Kotesovec, Jul 06 2013