A358042 -id:A358042 - cp's OEIS frontend

A358159 a(n) is the permanent of the n X n matrix M(n) that is defined by M[i,j] = ij - floor(ij/3).

1, 1, 7, 102, 4396, 374216, 49857920, 11344877568, 3879729283968, 1804571320405248, 1195546731955854336, 1058730877124859138048, 1184751018265831288602624, 1725335046543668616765112320, 3147123030650561978295975936000, 6934187745940804400441946931200000, 18840570649600136750602236509552640000

Offset: 0

Stefano Spezia, Nov 01 2022

nonn

The matrix M(n) is the n-th principal submatrix of the rectangular array A143976 and it is singular for n > 3.

			a(5) = 374216:
    1   2   2   3   4
    2   3   4   6   7
    2   4   6   8  10
    3   6   8  11  14
    4   7  10  14  17

Cf. A143976.

Cf. A071619 (matrix element M[n,n]), A358042 (trace of M(n)), A358160 (hafnian of M(2*n)).

Join[{1},Table[Permanent[Table[i*j-Floor[i*j/3],{i,n},{j,n}]],{n,17}]]

from sympy import Matrix
def A358159(n): return Matrix(n,n,[i*j-i*j//3 for i in range(1,n+1) for j in range(1,n+1)]).per() if n else 1 # Chai Wah Wu, Nov 02 2022

A358160 a(n) is the hafnian of the 2n X 2n symmetric matrix defined by M[i,j] = ij - floor(ij/3).

Original entry on oeis.org

1, 2, 40, 3884, 1016376, 534983256, 510252517152, 802452895865280, 1901953775079849600, 6537796866589765507200, 31381746234057256630521600

Offset: 0

Stefano Spezia, Nov 01 2022

The matrix M(n) is the n-th principal submatrix of the rectangular array A143976.

			a(2) = 40:
    1   2   2   3
    2   3   4   6
    2   4   6   8
    3   6   8  11

Wikipedia, Hafnian
Wikipedia, Symmetric matrix

Cf. A143976.

Cf. A071619 (matrix element M[n,n]), A358159 (permanent of M(2*n)), A358042 (trace of M(n)).

M[i_, j_, n_]:=Part[Part[Table[r*c-Floor[r*c/3], {r, n}, {c, n}], i], j]; a[n_]:=Sum[Product[M[Part[PermutationList[s, 2n], 2i-1], Part[PermutationList[s, 2n], 2i], 2n], {i, n}], {s, SymmetricGroup[2n]//GroupElements}]/(n!*2^n); Array[a, 6, 0]

tm(n) = matrix(n, n, i, j, i*j - (i*j)\3);
a(n) = my(m = tm(2*n), s=0); forperm([1..2*n], p, s += prod(j=1, n, m[p[2*j-1], p[2*j]]); ); s/(n!*2^n); \\ Michel Marcus, May 02 2023

a(6) from Michel Marcus, May 02 2023

a(7)-a(10) from Pontus von Brömssen, Oct 15 2023

cp's OEIS Frontend

A358159 a(n) is the permanent of the n X n matrix M(n) that is defined by M[i,j] = ij - floor(ij/3).

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A358160 a(n) is the hafnian of the 2n X 2n symmetric matrix defined by M[i,j] = ij - floor(ij/3).

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cp's OEIS Frontend

A358159 a(n) is the permanent of the n X n matrix M(n) that is defined by M[i,j] = i*j - floor(i*j/3).

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Python

A358160 a(n) is the hafnian of the 2n X 2n symmetric matrix defined by M[i,j] = i*j - floor(i*j/3).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Extensions

A358159 a(n) is the permanent of the n X n matrix M(n) that is defined by M[i,j] = ij - floor(ij/3).

A358160 a(n) is the hafnian of the 2n X 2n symmetric matrix defined by M[i,j] = ij - floor(ij/3).