A085750 -id:A085750 - cp's OEIS frontend

A374279 a(n) is the minimal determinant of an n X n symmetric Toeplitz matrix having 0 on the main diagonal and all the integers 1, 2, ..., n-1 off-diagonal.

1, 0, -1, 4, -44, -946, -8281, -592100, -25369920, -511563816, -55400732937

Offset: 0

Stefano Spezia, Jul 02 2024

			a(5) = -946:
  [0, 1, 4, 2, 3]
  [1, 0, 1, 4, 2]
  [4, 1, 0, 1, 4]
  [2, 4, 1, 0, 1]
  [3, 2, 4, 1, 0]

Wikipedia, Toeplitz Matrix.

Cf. A085750, A358323.

Cf. A085807 (minimal permanent), A374280 (maximal), A374281 (maximal absolute value), A374282 (minimal nonzero absolute value), A374283 (maximal permanent).

a[0]=1; a[n_]:=Min[Table[Det[ToeplitzMatrix[Join[{0},Part[Permutations[Range[n-1]],i]]]],{i,(n-1)!}]]; Array[a,11,0]

A374280 a(n) is the maximal determinant of an n X n symmetric Toeplitz matrix having 0 on the main diagonal and all the integers 1, 2, ..., n-1 off-diagonal.

Original entry on oeis.org

1, 0, -1, 8, 28, 282, 27495, 581268, 17344692, 1246207300, 33366771123

Offset: 0

Stefano Spezia, Jul 02 2024

			a(5) = 282:
  [0, 3, 4, 2, 1]
  [3, 0, 3, 4, 2]
  [4, 3, 0, 3, 4]
  [2, 4, 3, 0, 3]
  [1, 2, 4, 3, 0]

Wikipedia, Toeplitz Matrix.

Cf. A085750, A358324.

Cf. A085807 (minimal permanent), A374279 (minimal), A374281 (maximal absolute value), A374282 (minimal nonzero absolute value), A374283 (maximal permanent).

a[0]=1; a[n_]:=Max[Table[Det[ToeplitzMatrix[Join[{0},Part[Permutations[Range[n-1]],i]]]],{i,(n-1)!}]]; Array[a,11,0]

A374281 a(n) is the maximal absolute value of the determinant of an n X n symmetric Toeplitz matrix having 0 on the main diagonal and all the integers 1, 2, ..., n-1 off-diagonal.

Original entry on oeis.org

1, 0, 1, 8, 44, 946, 27495, 592100, 25369920, 1246207300, 55400732937

Offset: 0

Stefano Spezia, Jul 02 2024

			a(5) = 946:
  [0, 1, 4, 2, 3]
  [1, 0, 1, 4, 2]
  [4, 1, 0, 1, 4]
  [2, 4, 1, 0, 1]
  [3, 2, 4, 1, 0]

Wikipedia, Toeplitz Matrix.

Cf. A085750, A351609.

Cf. A085807 (minimal permanent), A374279 (minimal), A374280 (maximal), A374282 (minimal nonzero absolute value), A374283 (maximal permanent).

a[0]=1; a[n_]:=Max[Table[Abs[Det[ToeplitzMatrix[Join[{0},Part[Permutations[Range[n-1]],i]]]]],{i,(n-1)!}]]; Array[a,11,0]

a(n) = max(abs(A374279(n)), A374280(n)).

A374282 a(n) is the minimal absolute value of the determinant of a nonsingular n X n symmetric Toeplitz matrix having 0 on the main diagonal and all the integers 1, 2, ..., n-1 off-diagonal.

Original entry on oeis.org

1, 4, 12, 2, 13, 16, 21, 4, 1

Offset: 2

Stefano Spezia, Jul 02 2024

The offset is 2 because for n = 1 the matrix is null, and hence, singular.

			a(5) = 2:
  [0, 4, 1, 2, 3]
  [4, 0, 4, 1, 2]
  [1, 4, 0, 4, 1]
  [2, 1, 4, 0, 4]
  [3, 2, 1, 4, 0]

Wikipedia, Toeplitz Matrix.

Cf. A085750, A358325.

Cf. A085807 (minimal permanent), A374279 (minimal signed), A374280 (maximal signed), A374281 (maximal absolute value), A374283 (maximal permanent).

a[n_]:=Min[Select[Table[Abs[Det[ToeplitzMatrix[Join[{0},Part[Permutations[Range[n-1]],i]]]]],{i,(n-1)!}],Positive]]; Array[a,9,2]

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

A353452 a(n) is the determinant of the n X n symmetric matrix M(n) that is defined as M[i,j] = abs(i - j) if min(i, j) < max(i, j) <= 2*min(i, j), and otherwise 0.

Original entry on oeis.org

1, 0, -1, 0, 1, -4, 12, 64, -172, -1348, 3456, 34240, -87084, 370640, -872336, -22639616, 52307088, -181323568, 399580288, 23627011200, -51305628400, -686160247552, 1545932859328, 68098264912128, -155370174372864, 6326621032802304, -13829529077133312, -1087288396552040448

Offset: 0

Stefano Spezia, Apr 19 2022

sign

			a(8) = -172:
    0,  1,  0,  0,  0,  0,  0,  0;
    1,  0,  1,  2,  0,  0,  0,  0;
    0,  1,  0,  1,  2,  3,  0,  0;
    0,  2,  1,  0,  1,  2,  3,  4;
    0,  0,  2,  1,  0,  1,  2,  3;
    0,  0,  3,  2,  1,  0,  1,  2;
    0,  0,  0,  3,  2,  1,  0,  1;
    0,  0,  0,  4,  3,  2,  1,  0.

Cf. A085750, A085807.

Cf. A000982 (number of zero matrix elements), A003983, A006918, A007590 (number of positive matrix elements), A049581, A051125, A173997, A350050, A352967, A353453 (permanent).

Join[{1},Table[Det[Table[If[Min[i,j]

a(n) = matdet(matrix(n, n, i, j, if ((min(i,j) < max(i,j)) && (max(i,j) <= 2*min(i,j)), abs(i-j)))); \\ Michel Marcus, Apr 20 2022

from sympy import Matrix
def A353452(n): return Matrix(n, n, lambda i, j: abs(i-j) if min(i,j)Chai Wah Wu, Aug 29 2023

Sum_{i=1..n+1-k} M[i,i+k] = A173997(n, k) with 1 <= k <= floor((n + 1)/2).
Sum_{i=1..n} Sum_{j=1..n} M[i,j] = 2*A006918(n-1).
Sum_{i=1..n} Sum_{j=1..n} M[i,j]^2 = A350050(n+1).

A353453 a(n) is the permanent of the n X n symmetric matrix M(n) that is defined as M[i,j] = abs(i - j) if min(i, j) < max(i, j) <= 2*min(i, j), and otherwise 0.

Original entry on oeis.org

1, 0, 1, 0, 1, 4, 64, 576, 7844, 63524, 882772, 11713408, 252996564, 5879980400, 184839020672, 5698866739200, 229815005974352, 9350598794677712, 480306381374466176, 23741710999960266176, 1446802666239931811472, 86153125248221968292928, 6197781268948296566634304

Offset: 0

Stefano Spezia, Apr 19 2022

nonn

			a(8) = 7844:
    0,  1,  0,  0,  0,  0,  0,  0;
    1,  0,  1,  2,  0,  0,  0,  0;
    0,  1,  0,  1,  2,  3,  0,  0;
    0,  2,  1,  0,  1,  2,  3,  4;
    0,  0,  2,  1,  0,  1,  2,  3;
    0,  0,  3,  2,  1,  0,  1,  2;
    0,  0,  0,  3,  2,  1,  0,  1;
    0,  0,  0,  4,  3,  2,  1,  0.

Cf. A085750, A085807.

Cf. A000982 (number of zero matrix elements), A003983, A006918, A007590 (number of positive matrix elements), A049581, A051125, A173997, A350050, A352967, A353452 (determinant).

Join[{1},Table[Permanent[Table[If[Min[i,j]

a(n) = matpermanent(matrix(n, n, i, j, if ((min(i,j) < max(i,j)) && (max(i,j) <= 2*min(i,j)), abs(i-j)))); \\ Michel Marcus, Apr 20 2022

from sympy import Matrix
def A353453(n): return Matrix(n, n, lambda i, j: abs(i-j) if min(i,j)Chai Wah Wu, Aug 29 2023

Sum_{i=1..n+1-k} M[i,i+k] = A173997(n, k) with 1 <= k <= floor((n + 1)/2).
Sum_{i=1..n} Sum_{j=1..n} M[i,j] = 2*A006918(n-1).
Sum_{i=1..n} Sum_{j=1..n} M[i,j]^2 = A350050(n+1).

A278857 a(n) = permanent M_n where M_n is the n X n matrix m(i,j) = (i-j)^2.

Original entry on oeis.org

1, 0, 1, 8, 676, 49600, 10335908, 2658757248, 1214367336000, 730771063280640, 642638269862752320, 736176718456263406080, 1122592471007868379259136, 2168016139899273930219233280, 5288852927890824307509101287680, 15889369670472598370104100032512000

Offset: 0

Vaclav Kotesovec, Nov 29 2016

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..37

Cf. A085750, A085807, A204249, A278845, A278847, A278858.

with(LinearAlgebra):
a:= n-> `if`(n=0, 1, Permanent(Matrix(n, (i, j)-> (i-j)^2))):
seq(a(n), n=0..16);  # Vaclav Kotesovec, Nov 30 2016, after Alois P. Heinz

Flatten[{1, Table[Permanent[Table[(i-j)^2, {i, 1, n}, {j, 1, n}]], {n, 1, 15}]}]

{a(n) = matpermanent(matrix(n, n, i, j, (i-j)^2))}
for(n=0, 20, print1(a(n), ", ")) \\ Vaclav Kotesovec, Aug 12 2021

A278858 a(n) = permanent M_n where M_n is the n X n matrix m(i,j) = abs(i^2-j^2).

Original entry on oeis.org

1, 0, 9, 240, 36864, 7741440, 3363235524, 2203143038208, 2248347011420160, 3260265586467690240, 6578570637254005920000, 17755898734939822501524480, 62673017366111480630785474560, 282641923592380319367599892725760, 1599753679036773033206787507696238848

Offset: 0

Vaclav Kotesovec, Nov 29 2016

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..36

Cf. A085750, A085807, A204249, A278845, A278847, A278857.

Flatten[{1, Table[Permanent[Table[Abs[i^2-j^2], {i, 1, n}, {j, 1, n}]], {n, 1, 15}]}]

{a(n) = matpermanent(matrix(n, n, i, j, abs(i^2-j^2)))}
for(n=0, 20, print1(a(n), ", ")) \\ Vaclav Kotesovec, Aug 12 2021

A085799 Determinant of the symmetric n X n matrix A defined by A[i,j] = abs(i^2 - j^2) for 1 <= i,j <= n.

Original entry on oeis.org

0, -9, 240, -6300, 181440, -5821200, 207567360, -8172964800, 352864512000, -16593453676800, 844757641728000, -46306798060723200, 2720119606364160000, -170493211041753600000, 11359219476176732160000, -801737767492652390400000, 59762476409805241712640000, -4691769415367001788620800000

Offset: 1

Yuval Dekel (dekelyuval(AT)hotmail.com), Jul 24 2003

sign

			From _Klaus Brockhaus_, Apr 28 2010: (Start)
a(5) = determinant(A) = 181440 where A is the matrix
  [ 0  3  8 15 24]
  [ 3  0  5 12 21]
  [ 8  5  0  7 16]
  [15 12  7  0  9]
  [24 21 16  9  0] (End)

Vincenzo Librandi, Table of n, a(n) for n = 1..140

Cf. A085750.

[ Determinant( SymmetricMatrix( &cat[ [ Abs(i^2-j^2): j in [1..i] ]: i in [1..n] ] ) ): n in [1..15] ]; // Klaus Brockhaus, Apr 28 2010

(Conjectured to give the same sequence, apart from signs): a:=n->sum((count(Permutation(n*2-1),size=n+1)),j=0..n)/2: seq(a(n), n=1..16); # Zerinvary Lajos, May 03 2007

A[i_, j_] := Abs[i^2 - j^2]; a[n_] := Det[Table[A[i, j], {i, n}, {j, n}]]; Table[a[n], {n, 44}] (* José María Grau Ribas, Apr 17 2010 *)

a(n) = matdet(matrix(n, n, i, j, abs(i^2-j^2))); \\ Michel Marcus, Aug 14 2017

From Vaclav Kotesovec, Jan 08 2019: (Start)
a(n) ~ -(-1)^n * 2^(2*n - 3/2) * n^(n+2) / exp(n).
Recurrence: (14*n - 27)*a(n) =  -8*(n-1)*(7*n + 4)*a(n-1) - 36*(2*n - 3)*a(n-2).
(End)

More terms from José María Grau Ribas, Apr 17 2010
Edited by N. J. A. Sloane, Apr 21 2010 at the suggestion of R. J. Mathar
More terms from Michel Marcus, Aug 14 2017

cp's OEIS Frontend

A374279 a(n) is the minimal determinant of an n X n symmetric Toeplitz matrix having 0 on the main diagonal and all the integers 1, 2, ..., n-1 off-diagonal.

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A374280 a(n) is the maximal determinant of an n X n symmetric Toeplitz matrix having 0 on the main diagonal and all the integers 1, 2, ..., n-1 off-diagonal.

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A374281 a(n) is the maximal absolute value of the determinant of an n X n symmetric Toeplitz matrix having 0 on the main diagonal and all the integers 1, 2, ..., n-1 off-diagonal.

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A374282 a(n) is the minimal absolute value of the determinant of a nonsingular n X n symmetric Toeplitz matrix having 0 on the main diagonal and all the integers 1, 2, ..., n-1 off-diagonal.

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

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A353452 a(n) is the determinant of the n X n symmetric matrix M(n) that is defined as M[i,j] = abs(i - j) if min(i, j) < max(i, j) <= 2*min(i, j), and otherwise 0.

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A353453 a(n) is the permanent of the n X n symmetric matrix M(n) that is defined as M[i,j] = abs(i - j) if min(i, j) < max(i, j) <= 2*min(i, j), and otherwise 0.

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A278857 a(n) = permanent M_n where M_n is the n X n matrix m(i,j) = (i-j)^2.

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