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a(n) is the hafnian of the 2n X 2n symmetric matrix defined by M[i,j] = i*j - ceiling(i*j/3).
1

%I #15 Oct 15 2023 09:27:07

%S 1,1,26,2704,698568,384890688,378771904512,597991783196160,

%T 1450380828625459200,5077825865646165964800,24487520383436615392204800

%N a(n) is the hafnian of the 2n X 2n symmetric matrix defined by M[i,j] = i*j - ceiling(i*j/3).

%C The matrix M(n) is the n-th principal submatrix of the rectangular array A143979.

%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) = 26:

%e 0 1 2 2

%e 1 2 4 5

%e 2 4 6 8

%e 2 5 8 10

%t M[i_, j_, n_]:=Part[Part[Table[r*c-Ceiling[r*c/3], {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 - ceil((i*j)/3));

%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. A143979.

%Y Cf. A030511 (matrix element M[n-1,n-1]), A358163 (permanent of M(n)).

%K nonn,hard,more

%O 0,3

%A _Stefano Spezia_, Nov 01 2022

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

%E a(7)-a(10) from _Pontus von Brömssen_, Oct 15 2023