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

0,2

Links

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

Wikipedia, Hafnian

Wikipedia, Symmetric matrix

Wikipedia, Toeplitz Matrix

Example

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.

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]

Prog

(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

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.

Sequence in context: A226057 A158886 A092957 * A171107 A218768 A195736

Adjacent sequences: A356478 A356479 A356480 * A356482 A356483 A356484

Keyword

nonn,hard,more

Author

Stefano Spezia, Aug 09 2022

Extensions

a(6) from Michel Marcus, May 02 2023

a(7)-a(10) from Pontus von Brömssen, Oct 14 2023

Status

approved