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A357503 a(n) is the hafnian of the 2n X 2n symmetric matrix whose element (i,j) equals abs(i-j).

%I A357503 #15 Oct 15 2023 09:26:44
%S A357503 1,1,8,174,7360,512720,53245824,7713320944,1486382446592,
%T A357503 367691598791424,113570289012090880
%N A357503 a(n) is the hafnian of the 2n X 2n symmetric matrix whose element (i,j) equals abs(i-j).
%H A357503 Wikipedia, <a href="https://en.wikipedia.org/wiki/Hafnian">Hafnian</a>
%H A357503 Wikipedia, <a href="https://en.wikipedia.org/wiki/Symmetric_matrix">Symmetric matrix</a>
%e A357503 a(2) = M_{1,2}*M_{3,4} + M_{1,3}*M_{2,4} + M_{1,4}*M_{2,3} = 8 is the hafnian of
%e A357503     0, 1, 2, 3;
%e A357503     1, 0, 1, 2;
%e A357503     2, 1, 0, 1;
%e A357503     3, 2, 1, 0.
%t A357503 M[i_, j_, n_]:=Part[Part[Table[Abs[r-c], {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 A357503 (PARI) tm(n) = matrix(n, n, i, j, abs(i-j));
%o A357503 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 A357503 Cf. A049581, A085750 (determinant of M(n)), A085807 (permanent of M(n)), A094053 (super- and subdiagonal sums of M(n) in reversed order), A144216 (row- and column sums of M(n)), A338456.
%K A357503 nonn,hard,more
%O A357503 0,3
%A A357503 _Stefano Spezia_, Oct 01 2022
%E A357503 a(6) from _Michel Marcus_, May 02 2023
%E A357503 a(7)-a(10) from _Pontus von Brömssen_, Oct 15 2023