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%I #59 Oct 14 2023 15:38:12
%S 1,2,43,2610,312081,61825050,18318396195,7586241152490,
%T 4184711271725985,2965919152834367730,2626408950849351178875
%N a(n) is the hafnian of the 2n X 2n symmetric matrix defined by M[i, j] = i + j - 1.
%C 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.
%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>
%e a(2) = 43 because the hafnian of
%e 1 2 3 4
%e 2 3 4 5
%e 3 4 5 6
%e 4 5 6 7
%e equals M_{1,2}*M_{3,4} + M_{1,3}*M_{2,4} + M_{1,4}*M_{2,3} = 43.
%t 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]
%o (PARI) tm(n) = matrix(n, n, i, j, i+j-1);
%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. 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)).
%Y Cf. A202038, A336114, A336286, A336400, A338456.
%Y Cf. A356481, A356482, A356483, A356484.
%K nonn,hard,more
%O 0,2
%A _Stefano Spezia_, Sep 25 2022
%E a(6) from _Michel Marcus_, May 02 2023
%E a(7)-a(10) from _Pontus von Brömssen_, Oct 14 2023
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