A166445 -id:A166445 - cp's OEIS frontend

A374139 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.

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

Offset: 0

Stefano Spezia, Jun 28 2024

A minor variant of A166445. - R. J. Mathar, Jul 01 2024

			a(4) = 1:
  [1, 1, 2, 3]
  [1, 1, 1, 2]
  [2, 1, 1, 1]
  [3, 2, 1, 1]

Max Alekseyev, Determinant of a certain Toeplitz matrix, MathOverflow, 2020.
Index entries for linear recurrences with constant coefficients, signature (1,-2,2,-1,1).

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

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

a(n) = matdet(matrix(n, n, i, j, if (i==j, 1, abs(i-j)))); \\ Michel Marcus, Jun 29 2024

from sympy import Matrix
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

G.f.: (1 + x^2 - x^3 + x^4)/((1 - x)*(1 + x^2)^2).
a(n) = a(n-1) - 2*a(n-2) + 2*a(n-3) - a(n-4) + a(n-5) for n > 4.
a(n) = (1 + A056594(n) + n*A056594(n+1))/2.
E.g.f.: (exp(x) + (1 + x)*cos(x))/2.
For a proof of the generating function and the recursion formula, see MathOverflow link. - Sela Fried, Jul 09 2024

A203993 Array: row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of {|i-j}, (A049581).

Original entry on oeis.org

0, -1, -1, 0, 1, 4, 6, 0, -1, -12, -32, -20, 0, 1, 32, 120, 140, 50, 0, -1, -80, -384, -648, -448, -105, 0, 1, 192, 1120, 2464, 2520, 1176, 196, 0, -1, -448, -3072, -8320, -11264, -7920, -2688, -336, 0, 1, 1024, 8064, 25920, 43680

Offset: 1

Clark Kimberling, Jan 09 2012

Let p(n)=p(n,x) be the characteristic polynomial of the n-th principal submatrix.  The zeros of p(n) are real, and they interlace the zeros of p(n+1).  See A202605 for a guide to related sequences.
Also the coefficients of the detour and distance polynomials of the n-path graph P_n. - Eric W. Weisstein, Apr 07 2017
p(n,x) = (-x)^n*(x*(1 + T(n, 1+1/x)) - n*S(n-1, 2*(1+1/x)))/(2*x), with the Chebyshev polynomials S (A049310) and T (A053120). This is the rewritten formula given below in the Mathematica program by Weisstein. - Wolfdieter Lang, Feb 02 2018

			The array T (a table if row n=0 is by convention put to 0) begins:
n\k     0      1      2       3       4       5      6      7     8    9  10 ...
(0:     0)
1:      0     -1
2:     -1      0      1
3:      4      6      0      -1
4:    -12    -32    -20       0       1
5:     32    120    140      50       0      -1
6:    -80   -384   -648    -448    -105       0      1
7:    192   1120   2464    2520    1176     196      0     -1
8:   -448  -3072  -8320  -11264   -7920   -2688   -336      0     1
9:   1024   8064  25920   43680   41184   21384   5544    540     0   -1
10: -2304 -20480 -76160 -153600 -182000 -128128 -51480 -10560  -825    0   1
... reformatted and extended. - _Wolfdieter Lang_, Feb 02 2018

(For references regarding interlacing roots, see A202605.)

Eric Weisstein's World of Mathematics, Detour Polynomial
Eric Weisstein's World of Mathematics, Distance Polynomial
Eric Weisstein's World of Mathematics, Path Graph
Index entries for sequences related to Chebyshev polynomials.

Cf. A049310, A049581, A053120, A085750 (column k=0, Det(M_n)), A166445(n-1) (alternating row sums), A202605.

(* begin*)
f[i_, j_] := Abs[i - j];
m[n_] := Table[f[i, j], {i, 1, n}, {j, 1, n}]
TableForm[m[6]] (* 6x6 principal submatrix *)
Flatten[Table[f[i, n + 1 - i],
{n, 1, 12}, {i, 1, n}]]  (* A049581 *)
p[n_] := CharacteristicPolynomial[m[n], x];
c[n_] := CoefficientList[p[n], x]
TableForm[Flatten[Table[p[n], {n, 1, 10}]]]
Table[c[n], {n, 1, 12}]
Flatten[%]    (* A203993 *)
TableForm[Table[c[n], {n, 1, 10}]]
(* end *)
CoefficientList[Table[CharacteristicPolynomial[SparseArray[{i_, j_} :> Abs[i - j], n], x], {n, 10}], x] //Flatten (* Eric W. Weisstein, Apr 07 2017 *)
CoefficientList[Table[((-x)^n (x + x ChebyshevT[2 n, Sqrt[1 + 1/(2 x)]] - n ChebyshevU[n - 1, 1 + 1/x]))/(2 x), {n, 10}], x] // Flatten (* Eric W. Weisstein, Apr 07 2017 *)
CoefficientList[Table[1/4 (2 (-x)^n + (-1 - x - Sqrt[1 + 2 x])^n + (-1 - x + Sqrt[1 + 2 x])^n + (n (-(-1 - x - Sqrt[1 + 2 x])^n + (-1 - x + Sqrt[1 + 2 x])^n))/Sqrt[1 + 2 x]), {n, 10}], x] // Flatten (* Eric W. Weisstein, Apr 07 2017 *)
CoefficientList[LinearRecurrence[{-4 - 5 x, -2 (2 + 6 x + 5 x^2), -2 x (2 + 6 x + 5 x^2), -x^3 (4 + 5 x), -x^5}, {-x, (-1 + x) (1 + x), -(2 + x) (-2 - 2 x + x^2), (-6 - 4 x + x^2) (2 + 4 x + x^2), -(4 + 6 x + x^2) (-8 - 18 x - 6 x^2 + x^3)}, 10], x] // Flatten (* Eric W. Weisstein, Apr 07 2017 *)

T(n, k) = [x^k] p(n,x), with p(n,x) = Determinant(M_n - x*1_n), with the n x n matrix M_n with entries M_n(i, j) = |i-j|, for n >= 1, k = 0, 1, ..., n. For p(n,x) see a comment above and the Mathematica formulas by Weisstein.- Wolfdieter Lang, Feb 02 2018

cp's OEIS Frontend

A374139 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.

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