A054516 Equivalent of the Kurepa hypothesis for left factorial.

0, 2, 2, 6, -4, 50, -258, 1862, -14824, 133506, -1334950, 14684582, -176214828, 2290792946, -32071101034, 481066515750, -7697064251728, 130850092279682, -2355301661033934, 44750731559645126, -895014631192902100, 18795307255050944562, -413496759611120779858

Offset: 3

Author

Aleksandar Petojevic, Apr 09 2000

Keywords

• sign
• easy

Links

• G. C. Greubel, Table of n, a(n) for n = 3..450
• Romeo Mestrovic, Variations of Kurepa's left factorial hypothesis, 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.
• A. Petojevic, M. Zizovic, Trees and the Kurepa hypothesis for left factorial, Filomat (Nis), (1999), 31-40.

Programs

Magma

[n eq 3 select 0 else -(n-3)*Self(n-3) + (n-2)*(n-3): n in [3..30]]; // G. C. Greubel, Mar 30 2019

Mathematica

(* Assuming offset 0 *)
Table[(-1)^n*n*((-1)^n - Subfactorial[n - 1]), {n,0,20}] (* Peter Luschny, Dec 30 2016 *)
RecurrenceTable[{a[n]+(n-3)*a[n-1]==(n-2)*(n-3), a[3]==0}, a, {n,3,30}] (* G. C. Greubel, Mar 30 2019 *)

PARI

m=30; v=concat([0], vector(m-1)); for(n=2, m, v[n]=-(n-1)*v[n-1] + n*(n-1)); v \\ G. C. Greubel, Mar 30 2019

Sage

@CachedFunction
def Self(n):
   if n == 3 : return 0
   return -(n-3)*Self(n-1) + (n-2)*(n-3)
[Self(n) for n in (3..30)] # G. C. Greubel, Mar 30 2019

Formula

a(3) = 0, a(n) = -(n-3)*a(n-1) + (n-3)*(n-2).

Conjecture: (-n+4)*a(n) + (-n^2+8*n-14)*a(n-1) + (n-2)*(n-4)*a(n-2) = 0. - R. J. Mathar, Jan 31 2014

Extensions

More terms from James Sellers, Apr 09 2000