A356481 a(n) is the hafnian of a symmetric Toeplitz matrix M(2*n) whose first row consists of 1, 2, ..., 2*n.

1, 2, 21, 532, 24845, 1856094, 203076097, 30633787976, 6097546660185, 1548899852221210, 489114616743840461

Offset: 0

Author

Stefano Spezia, Aug 09 2022

Keywords

• nonn
• hard
• more

Examples

			a(2) = 21 because the hafnian of
    1  2  3  4
    2  1  2  3
    3  2  1  2
    4  3  2  1
equals M_{1,2}*M_{3,4} + M_{1,3}*M_{2,4} + M_{1,4}*M_{2,3} = 21.
		

Links

• Wikipedia, Hafnian
• Wikipedia, Symmetric matrix
• Wikipedia, Toeplitz Matrix

Crossrefs

Cf. A001792 (absolute value of the determinant of M(n)), A204235 (permanent of M(n)).

Cf. A202038, A336114, A336286, A336400, A338456.

Cf. A356482, A356483, A356484.

Programs

Mathematica

k[i_]:=i; M[i_, j_, n_]:=Part[Part[ToeplitzMatrix[Array[k, 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) = my(m = matrix(n, n, i, j, if (i==1, j, if (j==1, i)))); for (i=2, n, for (j=2, n, m[i, j] = m[i-1, j-1]; ); ); m;
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 14 2023