A204249 -id:A204249 - cp's OEIS frontend

A278300 Decimal expansion of a constant related to the asymptotics of A204249.

2, 4, 5, 5, 4, 0, 7, 4, 8, 2, 2, 8, 4, 1, 2, 7, 9, 4, 9, 3, 7, 5, 7, 6, 7, 4, 0, 2, 6, 2, 0, 1, 7, 6, 0, 9, 8, 9, 4, 9, 3, 5, 2, 6, 4, 0, 8, 8, 3, 9, 2, 3, 5, 8, 8, 0, 6, 9, 7, 0, 5, 6, 0, 1, 1, 2, 8, 2, 0, 8, 9, 3, 8, 9, 3, 9, 9, 7, 3, 9, 8, 6, 4, 7, 5, 9, 0, 9, 8, 7, 0, 7, 6, 7, 7, 7, 2, 3, 0, 7, 1, 2, 5, 2, 6

Offset: 1

Vaclav Kotesovec, Dec 01 2016

			2.4554074822841279493757674026201760989493526408839235880697056011282...

Steven R. Finch, Mathematical Constants II, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018, p. 187.

Cf. A204249.

RealDigits[-2*LambertW[-1, -1/(2*Exp[1/2])]^2 / (1 + 2*LambertW[-1, -1/(2*Exp[1/2])]), 10, 120][[1]] (* Vaclav Kotesovec, Jun 13 2021 *)

Equals limit n->infinity (A204249(n)/(n!)^2)^(1/n).

Equals -2*LambertW(-1, -1/(2*exp(1/2)))^2 / (1 + 2*LambertW(-1, -1/(2*exp(1/2)))). - Vaclav Kotesovec, Jun 13 2021

More digits from Vaclav Kotesovec, Jun 13 2021

A085750 Determinant 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

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

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...

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

A346934 Permanent of the 2n X 2n matrix with the (i,j)-entry i-j (i,j=1..2n).

Original entry on oeis.org

-1, 52, -18660, 24446016, -85000104000, 647188836814080, -9486416237249952000, 244072502056661870592000, -10282514440038927957603532800, 671904022157076034864609763328000, -65203712913305114275839483698454528000

Offset: 1

Zhi-Wei Sun, Aug 08 2021

sign

The author has proved that a((p-1)/2) == 2 (mod p) for any odd prime p.
Conjecture 1: (-1)^n*a(n) > 0 for all n > 0. Also, a(n) == 0 (mod 2n+1) if 2n+1 is composite.
For any permutation f of {1,...,2n+1}, clearly Product_{j=1..2n+1} (j-f(j)) = -Product_{k=1..2n+1} (k-f^{-1}(k)). Thus the permanent of the matrix [i-j]_{1<=i,j<=2n+1} vanishes.
It is easy to see that per[i+j]{1<=i,j<n} = per[i+(n-j)]{1<=i,jA204249(2n) == a(n) (mod 2n+1).
Let D(2n) be the set of all derangements of {1,...,2n}. Clearly, a(n) is the sum of those Product_{j=1..2n}(j-f(j)) with f in the set D(2n).
Conjecture 2: For any odd prime p, the sum of those 1/Product_{j=1..p-1}(j-f(j)) with f in the set D(p-1) is congruent to (-1)^((p-1)/2) modulo p.

			a(1) is the permanent of the matrix [1-1,1-2;2-1,2-2] = [0,-1;1,0], which equals -1.

Vaclav Kotesovec, Table of n, a(n) for n = 1..19
Zhi-Wei Sun, Arithmetic properties of some permanents, arXiv:2108.07723 [math.GM], 2021.

Cf. A000166, A204249, A346949.

a[n_]:=a[n]=Permanent[Table[i-j,{i,1,2n},{j,1,2n}]];
Table[a[n],{n,1,11}]

a(n) = matpermanent(matrix(2*n, 2*n, i, j, i-j)); \\ Michel Marcus, Aug 08 2021

A278925 a(n) = permanent M_n where M_n is the n X n matrix m(i,j) = i^3 + j^3.

Original entry on oeis.org

1, 2, 113, 38736, 46311652, 143820883800, 966462062838180, 12412328008727861760, 278484670746890475310656, 10197331743850942940587152000, 577793817845799602600135280168000, 48534819511412868687827815575204633600, 5834998526939444017550860154062183732711680

Offset: 0

Vaclav Kotesovec, Dec 01 2016

nonn

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

Cf. A204249, A278847, A278926.

with(LinearAlgebra):
a:= n-> `if`(n=0, 1, Permanent(Matrix(n, (i, j)-> i^3+j^3))):
seq(a(n), n=0..16);

Flatten[{1, Table[Permanent[Table[i^3+j^3, {i, 1, n}, {j, 1, n}]], {n, 1, 15}]}]

{a(n) = matpermanent(matrix(n, n, i, j, i^3+j^3))}
for(n=0, 20, print1(a(n), ", ")) \\ Vaclav Kotesovec, Dec 21 2018

a(n) ~ c * d^n * n!^4 / n^(3/2), where d = 6.538385468679... and c = 0.84959670006...

A278926 a(n) = permanent M_n where M_n is the n X n matrix m(i,j) = i^4 + j^4.

Original entry on oeis.org

1, 2, 353, 561608, 4341274884, 111107400842568, 7493918659070379300, 1139021252689549522419840, 348457223545199873458486125120, 196982631587037086047232203674775680, 192443334239172066295878807351087122210880, 307899710379447999264505625949360598523097530880

Offset: 0

Vaclav Kotesovec, Dec 01 2016

nonn

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

Cf. A204249, A278847, A278925.

with(LinearAlgebra):
a:= n-> `if`(n=0, 1, Permanent(Matrix(n, (i, j)-> i^4+j^4))):
seq(a(n), n=0..16);

Flatten[{1, Table[Permanent[Table[i^4+j^4, {i, 1, n}, {j, 1, n}]], {n, 1, 15}]}]

{a(n) = matpermanent(matrix(n, n, i, j, i^4+j^4))}
for(n=0, 20, print1(a(n), ", ")) \\ Vaclav Kotesovec, Dec 21 2018

a(n) ~ c * d^n * n!^5 / n^2, where d = 11.83108... and c = 0.68284...

A278838 a(n) = det M_n where M_n is the n X n matrix m(i,j) = A000041(i+j).

Original entry on oeis.org

1, 2, 1, -2, 2, 3, 0, -3, -1, 4, -3, -3, 2, -1, -12, 12, 11, 6, -5, 0, 5, -4, -9, -11, 1, 4, -20, -20, -4, 9, -18, -27, 8, 52, -73, 83, 245, 88, -60, -217, -157, 74, -30, -99, 57, 74, -29, -36, 101, 320, -205, -206, 125, -109, -27, 139, -203, -644, -629, 723

Offset: 0

Vaclav Kotesovec, Nov 29 2016

sign

Vaclav Kotesovec, Table of n, a(n) for n = 0..500

Cf. A000041, A005249, A071543, A204249, A278839, A278840.

Flatten[{1, Table[Det[Table[PartitionsP[i+j], {i, n}, {j, n}]], {n, 1, 100}]}]

A278839 a(n) = det M_n where M_n is the n X n matrix m(i,j) = A000009(i+j).

Original entry on oeis.org

1, 1, -2, -1, 1, -1, 0, 1, 2, 1, -1, -1, -1, 3, -3, -7, -2, 3, -1, 0, 1, 1, -2, 2, 3, -2, 0, 0, -2, -3, -1, 0, 9, -5, -4, 0, 1, -1, -3, 1, 4, 3, 3, -7, -3, 3, 5, -48, 75, 143, 194, -272, 62, -31, -65, 46, 22, 3, -10, 2, 15, -15, -13, -2, 11, -1, -35, -26, 108

Offset: 0

Vaclav Kotesovec, Nov 29 2016

sign

Vaclav Kotesovec, Table of n, a(n) for n = 0..500

Cf. A000009, A005249, A071543, A204249, A278838, A278841.

Flatten[{1, Table[Det[Table[PartitionsQ[i+j], {i, n}, {j, n}]], {n, 1, 100}]}]

cp's OEIS Frontend

A278300 Decimal expansion of a constant related to the asymptotics of A204249.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

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

A085807 Permanent 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

PARI

Python

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

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

A346934 Permanent of the 2n X 2n matrix with the (i,j)-entry i-j (i,j=1..2n).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

A278925 a(n) = permanent M_n where M_n is the n X n matrix m(i,j) = i^3 + j^3.

Views

Author

Keywords

Links

Crossrefs

Programs