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A052169 Equivalent of the Kurepa hypothesis for left factorial. 5

%I

%S 1,2,5,19,91,531,3641,28673,254871,2523223,27526069,328018989,

%T 4239014627,59043418019,881715042417,14052333488521,238063061452591,

%U 4271909380510383,80941440893880941,1614781745832924773,33833522293642233339,742799603083145395579

%N Equivalent of the Kurepa hypothesis for left factorial.

%C a(n) = A002467(n)/(n-1) (A002467(n) = number of non-derangements of {1,2,...,n}). - _Emeric Deutsch_, Jun 15 2009

%H Alois P. Heinz, <a href="/A052169/b052169.txt">Table of n, a(n) for n = 2..450</a>

%H Juan S. Auli, <a href="https://search.proquest.com/openview/3f0cef1fbdb016d61e16412e4b855969/1">Pattern Avoidance in Inversion Sequences</a>, Ph. D. thesis, Dartmouth College, ProQuest Dissertations Publishing (2020), 27964164.

%H Juan S. Auli, Sergi Elizalde, <a href="https://arxiv.org/abs/1906.07365">Consecutive patterns in inversion sequences II: avoiding patterns of relations</a>, arXiv:1906.07365 [math.CO], 2019. See Table 1, p. 6.

%H Sergi Elizalde, <a href="https://arxiv.org/abs/2006.13842">Bijections for restricted inversion sequences and permutations with fixed points</a>, arXiv:2006.13842 [math.CO], 2020.

%H T. Kotek, J. A. Makowsky, <a href="http://arxiv.org/abs/1309.4020">Recurrence Relations for Graph Polynomials on Bi-iterative Families of Graphs</a>, arXiv preprint arXiv:1309.4020 [math.CO], 2013.

%H Romeo Mestrovic, <a href="http://arxiv.org/abs/1312.7037">Variations of Kurepa's left factorial hypothesis</a>, arXiv preprint arXiv:1312.7037 [math.NT], 2013.

%H Romeo Mestrovic, <a href="https://doi.org/10.2298/FIL1510207M">The Kurepa-Vandermonde matrices arising from Kurepa's left factorial hypothesis</a>, Filomat 29:10 (2015), 2207-2215; DOI 10.2298/FIL1510207M.

%H Alexsandar Petojevic, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL5/Petojevic/petojevic5.html">The Function vM_m(s; a; z) and Some Well-Known Sequences</a>, Journal of Integer Sequences, Vol. 5 (2002), Article 02.1.7.

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

%F 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

%F 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

%F From _Benedict W. J. Irwin_, Jun 02 2016: (Start)

%F Let y(-1)=1, y(0)=1,

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

%F a(n) = (n-2)!y(n-2).

%F (End)

%p 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

%p # second Maple program:

%p a:= proc(n) option remember; `if`(n<4, n-1,

%p (n-2)*a(n-1)+(n-3)*a(n-2))

%p end:

%p seq(a(n), n=2..25); # _Alois P. Heinz_, Aug 30 2016

%t Numerator[k=1; NestList[1+1/(k++ #1)&,1,12]] (* _Wouter Meeussen_, Mar 24 2007 *)

%t a[n_] := (n! - Subfactorial[n])/(n-1); Table[a[n], {n, 2, 23}] {* _Jean-Fran├žois Alcover_, Jul 21 2017, after _Emeric Deutsch_'s comment *)

%o (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

%Y Pairwise sums of A002467.

%K nonn,easy

%O 2,2

%A Aleksandar Petojevic (apetoje(AT)ptt.yu), Jan 26 2000

%E More terms from _James A. Sellers_, Jan 31 2000

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Last modified February 24 12:52 EST 2021. Contains 341569 sequences. (Running on oeis4.)