A052169 Equivalent of the Kurepa hypothesis for left factorial.

1, 2, 5, 19, 91, 531, 3641, 28673, 254871, 2523223, 27526069, 328018989, 4239014627, 59043418019, 881715042417, 14052333488521, 238063061452591, 4271909380510383, 80941440893880941, 1614781745832924773, 33833522293642233339, 742799603083145395579

Offset: 2

Author

Aleksandar Petojevic, Jan 26 2000

Keywords

• nonn
• easy

Links

• Alois P. Heinz, Table of n, a(n) for n = 2..450
• Juan S. Auli, Pattern Avoidance in Inversion Sequences, Ph. D. thesis, Dartmouth College, ProQuest Dissertations Publishing (2020), 27964164.
• Juan S. Auli and Sergi Elizalde, Consecutive patterns in inversion sequences II: avoiding patterns of relations, arXiv:1906.07365 [math.CO], 2019. See Table 1, p. 6.
• Sergi Elizalde, Bijections for restricted inversion sequences and permutations with fixed points, arXiv:2006.13842 [math.CO], 2020.
• T. Kotek and J. A. Makowsky, Recurrence Relations for Graph Polynomials on Bi-iterative Families of Graphs, arXiv preprint arXiv:1309.4020 [math.CO], 2013.
• Romeo Mestrovic, Variations of Kurepa's left factorial hypothesis, arXiv preprint arXiv:1312.7037 [math.NT], 2013.
• Romeo Mestrovic, The Kurepa-Vandermonde matrices arising from Kurepa's left factorial hypothesis, Filomat 29:10 (2015), 2207-2215; DOI 10.2298/FIL1510207M.
• Alexsandar Petojevic, The Function vM_m(s; a; z) and Some Well-Known Sequences, Journal of Integer Sequences, Vol. 5 (2002), Article 02.1.7.

Crossrefs

Pairwise sums of A002467.

Programs

Maple

a[2] := 1: a[3] := 2: for n from 4 to 21 do a[n] := (n-2)*a[n-1]+(n-3)*a[n-2] end do: seq(a[n], n = 2 .. 21); # Emeric Deutsch, Jun 15 2009
# second Maple program:
a:= proc(n) option remember; `if`(n<4, n-1,
      (n-2)*a(n-1)+(n-3)*a(n-2))
    end:
seq(a(n), n=2..25);  # Alois P. Heinz, Aug 30 2016

Mathematica

Numerator[k=1; NestList[1+1/(k++ #1)&,1,12]] (* Wouter Meeussen, Mar 24 2007 *)
a[n_] := (n! - Subfactorial[n])/(n-1); Table[a[n], {n, 2, 23}] (* Jean-François Alcover, Jul 21 2017, after Emeric Deutsch's comment *)

Sage

from sage.combinat.sloane_functions import ExtremesOfPermanentsSequence2 ; e = ExtremesOfPermanentsSequence2() ; it = e.gen(1,2,1) ; [next(it) for i in range(20)] #(5 rows) # Zerinvary Lajos, May 15 2009

Formula

a(2) = 1, a(3) = 2, a(n) = (n-2)*a(n-1) + (n-3)*a(n-2).

a(n) = A002467(n)/(n-1) (A002467(n) = number of non-derangements of {1,2,...,n}). - Emeric Deutsch, Jun 15 2009

a(n) = 2*floor((n+1)!*(n+3)/e+1/2) - 3*(floor(((n+1)!+1)/e)+ floor(((n+2)!+1)/e)) +(n+1)!+(n+2)!, n>1, with offset 0..a(0)= 1. - Gary Detlefs, Apr 18 2010

a(n) = 1/(n+1)*((n+2)!-floor(((n+2)!+1)/e)), with offset 0 a(n) = 1/(n-1)*(n! - floor((n!+1)/e)). - Gary Detlefs, Jul 11 2010

From Benedict W. J. Irwin, Jun 02 2016: (Start)

Let y(-1)=1, y(0)=1, and y(n) = (Sum_{k=0..n-1} y(k)+y(k-1))/n,

a(n) = (n-2)!*y(n-2).

(End)

a(n) = (Gamma(n+1,0)-exp(-1)*Gamma(n+1,-1))/(n-1). - Martin Clever, Mar 25 2023

Extensions

More terms from James Sellers, Jan 31 2000