%I #17 Nov 30 2017 03:55:42

%S 15,-47,197,-1029,6439,-46927,390249,-3645737,37792331,-430400211,

%T 5341017373,-71724018781,1036207207363983,-16024176975479,

%U 264083895859409,-4620276321889617,85520275455047059,-1669635965205539227,34287733935303686661

%N Kurepa determinant K_n.

%C This is the value of a certain determinant of order n-4 (see Metrovic 2013 for definition).

%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], 2013-2014.

%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.

%F Conjecture: a(n) ~ -(-1)^n * n! * exp(-1) / n^4. - _Vaclav Kotesovec_, Nov 30 2017

%p A236401 := proc(n)

%p local M,r,c ;

%p M := Matrix(n-4,n-4) ;

%p for r from 1 to n-4 do

%p for c from 1 to n-4 do

%p if r = 1 then

%p if c < n-4 then

%p M[r,c] := 1 ;

%p else

%p M[r,c] := 3 ;

%p end if;

%p elif r = n-4 then

%p if c = n-4 then

%p M[r,c] := -4 ;

%p elif c = n-5 then

%p M[r,c] := 1 ;

%p else

%p M[r,c] := 0 ;

%p end if;

%p elif c = n-4 then

%p M[r,c] := 2 ;

%p elif r > c+2 then

%p M[r,c] := 0 ;

%p elif r = c+2 then

%p M[r,c] := 1 ;

%p elif r = c+1 then

%p M[r,c] := r+1 ;

%p elif c = n-4 then

%p M[r,c] := 2 ;

%p else

%p M[r,c] := 1 ;

%p end if

%p end do:

%p end do:

%p LinearAlgebra[Determinant](M) ;

%p end proc:

%p seq(A236401(n),n=7..25) ;

%t A236401[n_] := Det[Table[Which[

%t r == 1, If[c < n - 4, 1, 3],

%t r == n - 4, Which[

%t c == n - 4, -4,

%t c == n - 5, 1,

%t True, 0],

%t c == n - 4, 2,

%t r > c + 2, 0,

%t r == c + 2, 1,

%t r == c + 1, r + 1,

%t c == n - 4, 2,

%t True, 1],

%t {r, 1, n - 4}, {c, 1, n - 4}]];

%t Table[A236401[n], {n, 7, 25}] (* _Jean-François Alcover_, Nov 30 2017, from Maple *)

%K sign

%O 7,1

%A _N. J. A. Sloane_, Jan 29 2014