A181286 -id:A181286 - cp's OEIS frontend

A358157 a(n) is the permanent of the n X n matrix M(n) that is defined by M[i,j] = floor(i*j/3).

1, 0, 0, 1, 32, 1422, 146720, 18258864, 3217515264, 910849979232, 316878962588928, 143616562358849280, 90359341652805156864, 68004478547050644357120, 63187026071337208000512000, 75392341069747600992153600000, 104962910849766568886449582080000, 174017685915978467201007058206720000

Offset: 0

Stefano Spezia, Nov 01 2022

nonn

The matrix M(n) is the n-th principal submatrix of the rectangular array A143974.

det(M(0)) = 1, det(M(3)) = -1, and otherwise det(M(n)) = 0.

			a(5) = 1422:
    0  0  1  1  1
    0  1  2  2  3
    1  2  3  4  5
    1  2  4  5  6
    1  3  5  6  8

Cf. A143974.

Cf. A000212 (matrix element M[n,n]), A181286 (trace of M(n)), A358158 (hafnian of M(2*n)).

a[n_]:=Permanent[Table[Floor[i j/3],{i,n},{j,n}]]; Join[{1},Array[a,17]]

from sympy import Matrix
def A358157(n): return Matrix(n,n,[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

A358158 a(n) is the hafnian of the 2n X 2n symmetric matrix defined by M[i,j] = floor(i*j/3).

Original entry on oeis.org

1, 0, 4, 238, 31992, 9390096, 4755878928, 3802500283680, 4720879431568800, 8379987002639042400, 20346893722025317036800

Offset: 0

Stefano Spezia, Nov 01 2022

The matrix M(n) is the n-th principal submatrix of the rectangular array A143974.

Wikipedia, Hafnian
Wikipedia, Symmetric matrix

Cf. A143974.

Cf. A000212 (matrix element M[n,n]), A181286 (trace of M(n)), A358157 (permanent of M(n)).

M[i_, j_, n_]:=Part[Part[Table[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)\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

A358157 a(n) is the permanent of the n X n matrix M(n) that is defined by M[i,j] = floor(i*j/3).

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