OFFSET
0,2
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.
LINKS
EXAMPLE
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.
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]
PROG
(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
CROSSREFS
KEYWORD
nonn,hard,more
AUTHOR
Stefano Spezia, Sep 25 2022
EXTENSIONS
a(6) from Michel Marcus, May 02 2023
a(7)-a(10) from Pontus von Brömssen, Oct 14 2023
STATUS
approved