A338456 a(n) is the hafnian of a symmetric Toeplitz matrix M(2n) whose first row consists of a single zero followed by successive positive integers repeated (A004526).

1, 1, 4, 45, 968, 34265, 1799748, 131572357, 12770710096, 1589142683313, 246658484353100

Offset: 0

Author

Stefano Spezia, Oct 28 2020

Keywords

• nonn
• hard
• more

Examples

			a(2) = 4 because the hafnian of
0  1  1  2
1  0  1  1
1  1  0  1
2  1  1  0
equals M_{1,2}*M_{3,4} + M_{1,3}*M_{2,4} + M_{1,4}*M_{2,3} = 4.
		

Links

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

Crossrefs

Cf. A004526.

Cf. A002378 (conjectured determinant of M(2n+1)), A083392 (conjectured determinant of M(n+1)), A332566 (permanent of M(n)), A333119 (k-th super- and subdiagonal sums of the matrix M(n)).

Cf. A202038, A336114, A336286, A336400.

Programs

Mathematica

k[i_]:=Floor[i/2]; 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, 5, 0]

PARI

tm(n) = {my(m = matrix(n, n, i, j, if (i==1, j\2, if (j==1, i\2)))); 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, Nov 11 2020

Extensions

a(5) from Michel Marcus, Nov 11 2020

a(6)-a(10) from Pontus von Brömssen, Oct 14 2023