A356481 - OEIS

%I #18 Oct 14 2023 23:54:18

%S 1,2,21,532,24845,1856094,203076097,30633787976,6097546660185,

%T 1548899852221210,489114616743840461

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

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Hafnian">Hafnian</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Symmetric_matrix">Symmetric matrix</a>

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Toeplitz_matrix">Toeplitz Matrix</a>

%e a(2) = 21 because the hafnian of

%e     1  2  3  4

%e     2  1  2  3

%e     3  2  1  2

%e     4  3  2  1

%e equals M_{1,2}*M_{3,4} + M_{1,3}*M_{2,4} + M_{1,4}*M_{2,3} = 21.

%t 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]

%o (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;

%o 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

%Y Cf. A001792 (absolute value of the determinant of M(n)), A204235 (permanent of M(n)).

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

%Y Cf. A356482, A356483, A356484.

%K nonn,hard,more

%O 0,2

%A _Stefano Spezia_, Aug 09 2022

%E a(6) from _Michel Marcus_, May 02 2023

%E a(7)-a(10) from _Pontus von Brömssen_, Oct 14 2023