A353452 -id:A353452 - cp's OEIS frontend

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.

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).

A352967 Array read by antidiagonals: A(i, j) = abs(i - j) if min(i, j) < max(i, j) <= 2*min(i, j), and otherwise 0, with i >= 0 and j >= 0.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 2, 0, 2, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 2, 0, 2, 0, 0, 0, 0, 0, 0, 3, 1, 1, 3, 0, 0, 0, 0, 0, 0, 0, 2, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 3, 1, 1, 3, 0, 0, 0, 0, 0, 0, 0, 0, 4, 2, 0, 2, 4, 0, 0, 0, 0

Offset: 0

Stefano Spezia, Apr 21 2022

			The array begins:
    0, 0, 0, 0, 0, 0, 0, 0, ...
    0, 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, ...
    0, 0, 0, 2, 1, 0, 1, 2, ...
    0, 0, 0, 3, 2, 1, 0, 1, ...
    0, 0, 0, 0, 3, 2, 1, 0, ...
    ...

Cf. A003983, A049581, A051125, A307018 (antidiagonal half-sums), A353452, A353453.

Mathematica
```
A[i_,j_]:=If[Min[i, j]
				
```

A357420 a(n) is the hafnian of the 2n X 2n symmetric matrix defined by 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, 1, 1, 8, 86, 878, 13730, 348760, 11622396, 509566864, 26894616012, 1701189027944, 125492778658096, 10738546182981256, 1049631636279244832, 117756049412699967072

Offset: 0

Stefano Spezia, Sep 27 2022

			a(4) = 86:
    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.

Wikipedia, Hafnian
Wikipedia, Symmetric matrix

Cf. A000982 (number of zero matrix elements of M(n)), A003983, A007590 (number of positive matrix elements of M(n)), A049581, A051125, A352967, A353452 (determinant of M(n)), A353453 (permanent of M(n)).
Cf. A202038, A336114, A336286, A336400, A338456.
Cf. A356481, A356482, A356483, A356484, A357279.

M[i_, j_, n_] := If[Min[i, j] < Max[i, j] <= 2 Min[i, j], Abs[j - i], 0]; a[n_] := Sum[Product[M[Part[PermutationList[s, 2 n], 2 i - 1], Part[PermutationList[s, 2 n], 2 i], 2 n], {i, n}], {s, SymmetricGroup[2 n] // GroupElements}]/(n!*2^n); Array[a, 6, 0]

a(6)-a(15) from Pontus von Brömssen, Oct 16 2023

cp's OEIS Frontend

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.

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

PARI

Python

Formula

A352967 Array read by antidiagonals: A(i, j) = abs(i - j) if min(i, j) < max(i, j) <= 2*min(i, j), and otherwise 0, with i >= 0 and j >= 0.

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

A357420 a(n) is the hafnian of the 2n X 2n symmetric matrix defined by M[i,j] = abs(i - j) if min(i, j) < max(i, j) <= 2*min(i, j), and otherwise 0.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

Extensions