A358164 a(n) is the hafnian of the 2n X 2n symmetric matrix defined by M[i,j] = i*j - ceiling(i*j/3).

1, 1, 26, 2704, 698568, 384890688, 378771904512, 597991783196160, 1450380828625459200, 5077825865646165964800, 24487520383436615392204800

Offset: 0

Author

Stefano Spezia, Nov 01 2022

Keywords

• nonn
• hard
• more

Comments

The matrix M(n) is the n-th principal submatrix of the rectangular array A143979.

Examples

			a(2) = 26:
    0  1   2   2
    1  2   4   5
    2  4   6   8
    2  5   8  10
		

Links

• Wikipedia, Hafnian
• Wikipedia, Symmetric matrix

Crossrefs

Cf. A143979.

Cf. A030511 (matrix element M[n-1,n-1]), A358163 (permanent of M(n)).

Programs

Mathematica

M[i_, j_, n_]:=Part[Part[Table[r*c-Ceiling[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]

PARI

tm(n) = matrix(n, n, i, j, i*j - ceil((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

Extensions

a(6) from Michel Marcus, May 02 2023

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