A357279 a(n) is the hafnian of the 2n X 2n symmetric matrix defined by M[i, j] = i + j - 1.

1, 2, 43, 2610, 312081, 61825050, 18318396195, 7586241152490, 4184711271725985, 2965919152834367730, 2626408950849351178875

Offset: 0

Author

Stefano Spezia, Sep 25 2022

Keywords

• nonn
• hard
• more

Comments

The n X n matrix M is the n-th principal submatrix of A002024 considered as an array, and it is singular for n > 2.

Examples

			a(2) = 43 because the hafnian of
    1  2  3  4
    2  3  4  5
    3  4  5  6
    4  5  6  7
equals M_{1,2}*M_{3,4} + M_{1,3}*M_{2,4} + M_{1,4}*M_{2,3} = 43.
		

Links

• Wikipedia, Hafnian
• Wikipedia, Symmetric matrix

Crossrefs

Cf. A002024, A002415 (absolute value of the coefficient of x^(n-2) in the characteristic polynomial of M(n)), A095833 (k-th super- and subdiagonal sums of the matrix M(n)), A204248 (permanent of M(n)).

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

Cf. A356481, A356482, A356483, A356484.

Programs

Mathematica

M[i_, j_, n_]:=Part[Part[Table[r+c-1,{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, i+j-1);
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