A374139 - OEIS

%I #42 Jul 23 2024 20:33:12

%S 1,1,0,-1,1,3,0,-3,1,5,0,-5,1,7,0,-7,1,9,0,-9,1,11,0,-11,1,13,0,-13,1,

%T 15,0,-15,1,17,0,-17,1,19,0,-19,1,21,0,-21,1,23,0,-23,1,25,0,-25,1,27,

%U 0,-27,1,29,0,-29,1,31,0,-31,1,33,0,-33,1,35,0,-35,1,37,0,-37

%N a(n) is the determinant of the symmetric Toeplitz matrix of order n whose element (i,j) equals abs(i-j) or 1 if i = j.

%C A minor variant of A166445. - _R. J. Mathar_, Jul 01 2024

%H Max Alekseyev, <a href="https://mathoverflow.net/q/371595">Determinant of a certain Toeplitz matrix</a>, MathOverflow, 2020.

%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (1,-2,2,-1,1).

%F G.f.: (1 + x^2 - x^3 + x^4)/((1 - x)*(1 + x^2)^2).

%F a(n) = a(n-1) - 2*a(n-2) + 2*a(n-3) - a(n-4) + a(n-5) for n > 4.

%F a(n) = (1 + A056594(n) + n*A056594(n+1))/2.

%F E.g.f.: (exp(x) + (1 + x)*cos(x))/2.

%F For a proof of the generating function and the recursion formula, see MathOverflow link. - _Sela Fried_, Jul 09 2024

%e a(4) = 1:

%e   [1, 1, 2, 3]

%e   [1, 1, 1, 2]

%e   [2, 1, 1, 1]

%e   [3, 2, 1, 1]

%t a[n_]:=Det[Table[If[i == j, 1, Abs[i - j]], {i, n}, {j, n}]]; Join[{1}, Array[a, 75]]

%o (PARI) a(n) = matdet(matrix(n, n, i, j, if (i==j, 1, abs(i-j)))); \\ _Michel Marcus_, Jun 29 2024

%o (Python)

%o from sympy import Matrix

%o def A374139(n): return Matrix(n,n,[abs(j-k) if j!=k else 1 for j in range(n) for k in range(n)]).det() # _Chai Wah Wu_, Jul 01 2024

%Y Cf. A056594, A071078, A085750, A374140 (permanent).

%K sign,easy

%O 0,6

%A _Stefano Spezia_, Jun 28 2024