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

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

Offset

0,3

Comments

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

Links

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

Wikipedia, Hafnian

Wikipedia, Symmetric matrix

Example

a(2) = 4:
    0  0  1  1
    0  1  2  2
    1  2  3  4
    1  2  4  5

Mathematica

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]

Prog

(PARI) 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

Crossrefs

Cf. A143974.

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

Sequence in context: A281614 A132551 A333864 * A013953 A051753 A323996

Adjacent sequences: A358155 A358156 A358157 * A358159 A358160 A358161

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