A356483 a(n) is the hafnian of a symmetric Toeplitz matrix M(2*n) whose first row consists of prime(1), prime(2), ..., prime(2*n).

1, 3, 55, 2999, 347391, 69702479, 22441691645, 10776262328919, 7190279422736061, 6439969796874334809, 7447188585071730451961

Offset: 0

Author

Stefano Spezia, Aug 09 2022

Keywords

• nonn
• hard
• more

Examples

			a(2) = 55 because the hafnian of
    2  3  5  7
    3  2  3  5
    5  3  2  3
    7  5  3  2
equals M_{1,2}*M_{3,4} + M_{1,3}*M_{2,4} + M_{1,4}*M_{2,3} = 55.
		

Links

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

Crossrefs

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

Cf. A356481, A356482, A356484.

Cf. A356490 (determinant of M(n)), A356491 (permanent of M(n)).

Programs

Mathematica

k[i_]:=Prime[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, prime(j), if (j==1, prime(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