VOOZH about

URL: https://oeis.org/A052169

⇱ A052169 - OEIS

login
A052169
Equivalent of the Kurepa hypothesis for left factorial.
5
1, 2, 5, 19, 91, 531, 3641, 28673, 254871, 2523223, 27526069, 328018989, 4239014627, 59043418019, 881715042417, 14052333488521, 238063061452591, 4271909380510383, 80941440893880941, 1614781745832924773, 33833522293642233339, 742799603083145395579
OFFSET
2,2
LINKS
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.
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.
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
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
# Alternative:
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 *)
PROG
(SageMath) 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
CROSSREFS
Pairwise sums of A002467.
Sequence in context: A052324 A020115 A103816 * A020019 A020109 A391975
KEYWORD
nonn,easy
AUTHOR
EXTENSIONS
More terms from James Sellers, Jan 31 2000
STATUS
approved