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

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

Offset

0,2

Comments

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

Links

Table of n, a(n) for n=0..10.

Wikipedia, Hafnian

Wikipedia, Symmetric matrix

Example

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

Mathematica

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]

Prog

(PARI) 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

Crossrefs

Cf. A143976.

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

Sequence in context: A246742 A293950 A104134 * A328553 A162868 A059476

Adjacent sequences: A358157 A358158 A358159 * A358161 A358162 A358163

Keyword

nonn,hard,more

Author

Stefano Spezia, Nov 01 2022

Extensions

a(6) from Michel Marcus, May 02 2023

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

Status

approved