A357503 a(n) is the hafnian of the 2n X 2n symmetric matrix whose element (i,j) equals abs(i-j).

1, 1, 8, 174, 7360, 512720, 53245824, 7713320944, 1486382446592, 367691598791424, 113570289012090880

Offset: 0

Author

Stefano Spezia, Oct 01 2022

Keywords

• nonn
• hard
• more

Examples

			a(2) = M_{1,2}*M_{3,4} + M_{1,3}*M_{2,4} + M_{1,4}*M_{2,3} = 8 is the hafnian of
    0, 1, 2, 3;
    1, 0, 1, 2;
    2, 1, 0, 1;
    3, 2, 1, 0.
		

Links

• Wikipedia, Hafnian
• Wikipedia, Symmetric matrix

Crossrefs

Cf. A049581, A085750 (determinant of M(n)), A085807 (permanent of M(n)), A094053 (super- and subdiagonal sums of M(n) in reversed order), A144216 (row- and column sums of M(n)), A338456.

Programs

Mathematica

M[i_, j_, n_]:=Part[Part[Table[Abs[r-c], {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, abs(i-j));
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