A278845 -id:A278845 - 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

A085807 Permanent of the symmetric n X n matrix A defined by A[i,j] = |i-j| for 1 <= i,j <= n.

Original entry on oeis.org

1, 0, 1, 4, 64, 1152, 34372, 1335008, 69599744, 4577345152, 374491314176, 37154032517376, 4402467119882240, 613680867638476800, 99443966100565999872, 18534733913629064343552, 3937496200758879526977536, 945776134421421651222708224, 255043190756805184245158084608

Offset: 0

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

nonn

Conjecture: For any odd prime p, we have a(p) == -1/2 (mod p). - Zhi-Wei Sun, Aug 30 2021

Conjecture: a(n) is the minimal permanent of an n X n symmetric Toeplitz matrix having 0 on the main diagonal and all the integers 1, 2, ..., n-1 off-diagonal. - Stefano Spezia, Jul 05 2024

Vaclav Kotesovec, Table of n, a(n) for n = 0..37
Zhi-Wei Sun, Arithmetic properties of some permanents, arXiv:2108.07723 [math.GM], 2021.

Cf. A085750, A204249, A278845, A278847.

with(LinearAlgebra):
a:= n-> `if`(n=0, 1, Permanent(Matrix(n, (i, j)-> abs(i-j)))):
seq(a(n), n=0..18);  # Alois P. Heinz, Nov 14 2016

a[n_]:=Permanent[Table[Abs[i - j], {i, n}, {j, n}]]; Join[{1}, Array[a, 18]] (* Stefano Spezia, Jun 28 2024 *)

permRWNb(a)= n=matsize(a)[1]; if(n==1,return(a[1,1])); sg=1; in=vectorv(n); x=in; x=a[,n]-sum(j=1,n,a[,j])/2; p=prod(i=1,n,x[i]); for(k=1,2^(n-1)-1,sg=-sg; j=valuation(k,2)+1; z=1-2*in[j]; in[j]+=z; x+=z*a[,j]; p+=prod(i=1,n,x[i],sg)); return(2*(2*(n%2)-1)*p)
for(n=1,22,a=matrix(n,n,i,j,abs(i-j));print1(permRWNb(a)",")) \\  Herman Jamke (hermanjamke(AT)fastmail.fm), May 14 2007

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

from sympy import Matrix
def A085807(n): return Matrix(n,n,[abs(j-k) for j in range(n) for k in range(n)]).per() # Chai Wah Wu, Sep 14 2021

More terms from Vladeta Jovovic, Jul 26 2003

a(0)=1 prepended by Alois P. Heinz, Nov 14 2016

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...

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

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

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A085807 Permanent of the symmetric n X n matrix A defined by A[i,j] = |i-j| for 1 <= i,j <= n.

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

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

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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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Maple

Mathematica

PARI

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

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