A085807 -id:A085807 - cp's OEIS frontend

A085750 Determinant of the symmetric n X n matrix A defined by A[i,j] = |i-j| for 1 <= i,j <= n.

0, -1, 4, -12, 32, -80, 192, -448, 1024, -2304, 5120, -11264, 24576, -53248, 114688, -245760, 524288, -1114112, 2359296, -4980736, 10485760, -22020096, 46137344, -96468992, 201326592, -419430400, 872415232, -1811939328, 3758096384, -7784628224, 16106127360

Offset: 1

Yuval Dekel (dekelyuval(AT)hotmail.com), Jul 21 2003

The determinant of the distance matrix of a tree with vertex set {1,2,...,n}. The distance matrix is the n X n matrix in which the (i,j)-term is the number of edges in the unique path from vertex i to vertex j. [The matrix A in the definition is the distance matrix of the path-tree 1-2-...-n.]
Hankel transform of A100071. Also Hankel transform of C(2n-2,n-1)(-1)^(n-1). Inverse binomial transform of -n. - Paul Barry, Jan 11 2007
Pisano period lengths: 1, 1, 3, 1, 20, 3, 42, 1, 9, 20, 55, 3,156, 42, 60, 1,136, 9,171, 20, ... - R. J. Mathar, Aug 10 2012

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Emmanuel Briand, Luis Esquivias, Álvaro Gutiérrez, Adrián Lillo, and Mercedes Rosas, Determinant of the distance matrix of a tree, Proceedings of the 36th Conference on Formal Power Series and Algebraic Combinatorics (Bochum), Séminaire Lotharingien de Combinatoire 91B (2024) Article #29, 12 pp.
R. L. Graham and H. O. Pollak, On the addressing problem for loop switching, Bell System Tech. J., 50, 1971, 2495-2519.
Tanya Khovanova, Recursive Sequences
R. Merris, The distance spectrum of a tree, J. Graph Theory, 14, No. 3, 1990,365-369.
Index entries for linear recurrences with constant coefficients, signature (-4,-4).

Essentially the same as A001787.

Cf. A085807, A100071, A203993, A204249, A278845, A278847.

seq((-1)^(n-1)*(n-1)*2^(n-2), n = 1 .. 31);

Table[-(-1)^n*2^(n - 2)*(n - 1), {n, 1, 50}] (* Vladimir Joseph Stephan Orlovsky, Jun 10 2011 *)
LinearRecurrence[{-4,-4},{0,-1},40] (* Harvey P. Dale, Apr 14 2014 *)
CoefficientList[Series[-x/(1 + 2 x)^2, {x, 0, 40}], x] (* Vincenzo Librandi, Apr 15 2014 *)

a(n) = (-1)^n*(1-n)<<(n-2) \\ Charles R Greathouse IV, Sep 30 2022

a(n) = (-1)^(n+1) * (n-1) * 2^(n-2) = (-1)^(n+1) * A001787(n-1).
G.f.: -x/(1+2x)^2. - Paul Barry, Jan 11 2007
a(n) = -4*a(n-1) - 4*a(n-2); a(1) = 0, a(1) = -1. - Philippe Deléham, Nov 03 2008
E.g.f.: -x*exp(-2*x). - Stefano Spezia, Sep 30 2022

More terms from Philippe Deléham, Nov 16 2008

A204249 Permanent of the n-th principal submatrix of A003057.

Original entry on oeis.org

1, 2, 17, 336, 12052, 685080, 56658660, 6428352000, 958532774976, 181800011433600, 42745508545320000, 12203347213269273600, 4158410247782904833280, 1667267950805177583582720, 776990110000329481864608000, 416483579190482716042690560000

Offset: 0

Clark Kimberling, Jan 14 2012

nonn

I have proved that for any odd prime p we have a(p) == p (mod p^2). - Zhi-Wei Sun, Aug 30 2021

Vaclav Kotesovec, Table of n, a(n) for n = 0..36
Zhi-Wei Sun, Arithmetic properties of some permanents, arXiv:2108.07723 [math.GM], 2021.

Cf. A000442, A003057, A085750, A085807, A278845, A278847, A278300, A278925, A278926.

with(LinearAlgebra):
a:= n-> `if`(n=0, 1, Permanent(Matrix(n, (i, j)-> i+j))):
seq(a(n), n=0..16);  # Alois P. Heinz, Nov 14 2016

f[i_, j_] := i + j;
m[n_] := Table[f[i, j], {i, 1, n}, {j, 1, n}]
TableForm[m[8]] (* 8x8 principal submatrix *)
Flatten[Table[f[i, n + 1 - i],
  {n, 1, 12}, {i, 1, n}]]  (* A003057 *)
Permanent[m_] :=
  With[{a = Array[x, Length[m]]},
   Coefficient[Times @@ (m.a), Times @@ a]];
Table[Permanent[m[n]], {n, 1, 15}]  (* A204249 *)

{a(n) = matpermanent(matrix(n, n, i, j, i+j))}
for(n=0, 20, print1(a(n), ", ")) \\ Vaclav Kotesovec, Dec 21 2018

From Vaclav Kotesovec, Dec 01 2016: (Start)
a(n) ~ c * d^n * (n!)^2 / sqrt(n), where d = A278300 = 2.455407482284127949... and c = 1.41510164826...
a(n) ~ c * d^n * n^(2*n + 1/2), where d = A278300/exp(2) = 0.332303267076220516... and c = 8.89134588451...
(End)

a(0)=1 prepended and one more term added by Alois P. Heinz, Nov 14 2016

A278847 a(n) = permanent M_n where M_n is the n X n matrix m(i,j) = i^2 + j^2.

Original entry on oeis.org

1, 2, 41, 3176, 620964, 246796680, 174252885732, 199381727959680, 345875291854507584, 864860593764292790400, 2996169331694350840741440, 13929521390709644084719495680, 84659009841182126038701730464000, 658043094413184868424932006273344000

Offset: 0

Vaclav Kotesovec, Nov 29 2016

nonn

From Zhi-Wei Sun, Aug 19 2021: (Start)
I have proved that a(n) == (-1)^(n-1)*2*n! (mod 2n+1) whenever 2n+1 is prime.
Conjecture 1: If 2n+1 is composite, then a(n) == 0 (mod 2n+1).
Conjecture 2: If p = 4n+1 is prime, then the sum of those Product_{j=1..2n}(j^2-f(j)^2)^{-1} with f over all the derangements of {1,...,2n} is congruent to 1/(n!)^2 modulo p. (End)

Vaclav Kotesovec, Table of n, a(n) for n = 0..36
Zhi-Wei Sun, Arithmetic properties of some permanents, arXiv:2108.07723 [math.GM], 2021.

Cf. A005249, A085750, A085807, A204249, A278845, A278925, A278926, A346934, A346949.

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

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

a(n)={matpermanent(matrix(n, n, i, j, i^2 + j^2))} \\ Andrew Howroyd, Aug 21 2018

a(n) ~ c * d^n * (n!)^3 / n, where d = 3.809076776112918119... and c = 1.07739642254738...

A374068 a(n) is the permanent of the symmetric Toeplitz matrix of order n whose element (i,j) equals the |i-j|-th prime or 0 if i = j.

Original entry on oeis.org

1, 0, 4, 24, 529, 16100, 919037, 75568846, 9196890092, 1491628025318, 317579623173729, 86997150829931700, 29703399282858184713, 12512837775355494800500, 6397110844644502402189404, 3875565057688532269985283868, 2747710211567246171588232074225, 2265312860218073375019946448731300

Offset: 0

Stefano Spezia, Jun 27 2024

nonn

Conjecture: a(n) is the minimal permanent of an n X n symmetric Toeplitz matrix having 0 on the main diagonal and all the first n-1 primes off-diagonal. - Stefano Spezia, Jul 06 2024

			a(4) = 529:
  [0, 2, 3, 5]
  [2, 0, 2, 3]
  [3, 2, 0, 2]
  [5, 3, 2, 0]

Cf. A071079 (determinant), A085807, A306457, A318173.

Cf. A374067, A374069, A374070, A374071.

a[n_]:=Permanent[Table[If[i == j, 0, Prime[Abs[i - j]]], {i, 1, n}, {j, 1, n}]]; Join[{1},Array[a, 17]]

a(n) = matpermanent(matrix(n, n, i, j, if (i==j, 0, prime(abs(i-j))))); \\ Michel Marcus, Jun 28 2024

A278845 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, 4, 145, 19016, 6176676, 4038562000, 4664347807268, 8698721212922496, 24535712762777208384, 99585504924929052560640, 559305193643176161735904320, 4211594966980674975033969246720, 41428564066728305721531962537124096, 520897493876353116313789796095643304960

Offset: 0

Vaclav Kotesovec, Nov 29 2016

nonn

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

Cf. A005249, A085750, A085807, A204249, A278847.

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 29 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 09 2021

a(n) ~ c * d^n * (n!)^3 / n, where d = 6.14071825... and c = 1.79385445... - Vaclav Kotesovec, Aug 12 2021

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.

Original entry on oeis.org

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]

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).

cp's OEIS Frontend

A085750 Determinant of the symmetric n X n matrix A defined by A[i,j] = |i-j| for 1 <= i,j <= n.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

Extensions

A204249 Permanent of the n-th principal submatrix of A003057.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

Extensions

A278847 a(n) = permanent M_n where M_n is the n X n matrix m(i,j) = i^2 + j^2.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A374068 a(n) is the permanent of the symmetric Toeplitz matrix of order n whose element (i,j) equals the |i-j|-th prime or 0 if i = j.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

PARI

A278845 a(n) = permanent M_n where M_n is the n X n matrix m(i,j) = (i+j)^2.

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

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.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

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.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

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.

Views