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%I #13 Oct 15 2023 09:26:50
%S 1,0,4,238,31992,9390096,4755878928,3802500283680,4720879431568800,
%T 8379987002639042400,20346893722025317036800
%N a(n) is the hafnian of the 2n X 2n symmetric matrix defined by M[i,j] = floor(i*j/3).
%C The matrix M(n) is the n-th principal submatrix of the rectangular array A143974.
%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) = 4:
%e 0 0 1 1
%e 0 1 2 2
%e 1 2 3 4
%e 1 2 4 5
%t M[i_, j_, n_]:=Part[Part[Table[Floor[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)\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. A143974.
%Y Cf. A000212 (matrix element M[n,n]), A181286 (trace of M(n)), A358157 (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
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