A085807 -id:A085807

A085750

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

+10
24

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,3

COMMENTS

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

LINKS

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).

FORMULA

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

MAPLE

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

MATHEMATICA

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 *)

PROG

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

CROSSREFS

Essentially the same as A001787.

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

KEYWORD

easy,sign

AUTHOR

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

EXTENSIONS

More terms from Philippe Deléham, Nov 16 2008

STATUS

approved

A204249

Permanent of the n-th principal submatrix of A003057.

+10
18

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

COMMENTS

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

LINKS

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.

FORMULA

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)

MAPLE

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

MATHEMATICA

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 *)

PROG

(PARI) {a(n) = matpermanent(matrix(n, n, i, j, i+j))}

for(n=0, 20, print1(a(n), ", ")) \\ Vaclav Kotesovec, Dec 21 2018

CROSSREFS

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

KEYWORD

nonn

AUTHOR

Clark Kimberling, Jan 14 2012

EXTENSIONS

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

STATUS

approved

A278847

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

+10
10

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

COMMENTS

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)

LINKS

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.

FORMULA

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

MAPLE

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

MATHEMATICA

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

PROG

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

CROSSREFS

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

KEYWORD

nonn

AUTHOR

Vaclav Kotesovec, Nov 29 2016

STATUS

approved

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.

+10
9

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,3

COMMENTS

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

LINKS

Table of n, a(n) for n=0..17.

EXAMPLE

a(4) = 529:

[0, 2, 3, 5]

[2, 0, 2, 3]

[3, 2, 0, 2]

[5, 3, 2, 0]

MATHEMATICA

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

PROG

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

CROSSREFS

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

Cf. A374067, A374069, A374070, A374071.

KEYWORD

nonn

AUTHOR

Stefano Spezia, Jun 27 2024

STATUS

approved

A278845

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

+10
7

1, 4, 145, 19016, 6176676, 4038562000, 4664347807268, 8698721212922496, 24535712762777208384, 99585504924929052560640, 559305193643176161735904320, 4211594966980674975033969246720, 41428564066728305721531962537124096, 520897493876353116313789796095643304960

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

LINKS

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

FORMULA

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

MAPLE

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

MATHEMATICA

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

PROG

(PARI) {a(n) = matpermanent(matrix(n, n, i, j, (i+j)^2))}

for(n=0, 20, print1(a(n), ", ")) \\ Vaclav Kotesovec, Aug 09 2021

CROSSREFS

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

KEYWORD

nonn

AUTHOR

Vaclav Kotesovec, Nov 29 2016

STATUS

approved

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.

+10
5

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,4

LINKS

Table of n, a(n) for n=0..10.

Wikipedia, Toeplitz Matrix.

EXAMPLE

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]

MATHEMATICA

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

CROSSREFS

Cf. A085750, A358323.

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

KEYWORD

sign,hard,more

AUTHOR

Stefano Spezia, Jul 02 2024

STATUS

approved

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.

+10
5

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,4

LINKS

Table of n, a(n) for n=0..10.

Wikipedia, Toeplitz Matrix.

EXAMPLE

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]

MATHEMATICA

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

CROSSREFS

Cf. A085750, A358324.

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

KEYWORD

sign,hard,more

AUTHOR

Stefano Spezia, Jul 02 2024

STATUS

approved

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.

+10
5

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,4

LINKS

Table of n, a(n) for n=0..10.

Wikipedia, Toeplitz Matrix.

FORMULA

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

EXAMPLE

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]

MATHEMATICA

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]

CROSSREFS

Cf. A085750, A351609.

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

KEYWORD

nonn,hard,more

AUTHOR

Stefano Spezia, Jul 02 2024

STATUS

approved

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.

+10
5

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

2,2

COMMENTS

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

LINKS

Table of n, a(n) for n=2..10.

Wikipedia, Toeplitz Matrix.

EXAMPLE

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]

MATHEMATICA

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

CROSSREFS

Cf. A085750, A358325.

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

KEYWORD

nonn,hard,more

AUTHOR

Stefano Spezia, Jul 02 2024

STATUS

approved

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.

+10
4

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,6

LINKS

Table of n, a(n) for n=0..27.

FORMULA

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).

EXAMPLE

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.

MATHEMATICA

Join[{1}, Table[Det[Table[If[Min[i, j]<Max[i, j]<=2Min[i, j], Abs[j-i], 0], {i, n}, {j, n}]], {n, 27}]]

PROG

(PARI) 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

(Python)

from sympy import Matrix

def A353452(n): return Matrix(n, n, lambda i, j: abs(i-j) if min(i, j)<max(i, j)<=(min(i, j)<<1)+1 else 0).det() # Chai Wah Wu, Aug 29 2023

CROSSREFS

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).

KEYWORD

sign

AUTHOR

Stefano Spezia, Apr 19 2022

STATUS

approved