OFFSET
0,3
LINKS
EXAMPLE
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.
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]
PROG
(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
CROSSREFS
KEYWORD
nonn,hard,more
AUTHOR
Stefano Spezia, Oct 28 2020
EXTENSIONS
a(5) from Michel Marcus, Nov 11 2020
a(6)-a(10) from Pontus von Brömssen, Oct 14 2023
STATUS
approved