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A357279 a(n) is the hafnian of the 2n X 2n symmetric matrix defined by M[i, j] = i + j - 1.

%I A357279 #59 Oct 14 2023 15:38:12
%S A357279 1,2,43,2610,312081,61825050,18318396195,7586241152490,
%T A357279 4184711271725985,2965919152834367730,2626408950849351178875
%N A357279 a(n) is the hafnian of the 2n X 2n symmetric matrix defined by M[i, j] = i + j - 1.
%C A357279 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 A357279 Wikipedia, <a href="https://en.wikipedia.org/wiki/Hafnian">Hafnian</a>
%H A357279 Wikipedia, <a href="https://en.wikipedia.org/wiki/Symmetric_matrix">Symmetric matrix</a>
%e A357279 a(2) = 43 because the hafnian of
%e A357279     1  2  3  4
%e A357279     2  3  4  5
%e A357279     3  4  5  6
%e A357279     4  5  6  7
%e A357279 equals M_{1,2}*M_{3,4} + M_{1,3}*M_{2,4} + M_{1,4}*M_{2,3} = 43.
%t A357279 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 A357279 (PARI) tm(n) = matrix(n, n, i, j, i+j-1);
%o A357279 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 A357279 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 A357279 Cf. A202038, A336114, A336286, A336400, A338456.
%Y A357279 Cf. A356481, A356482, A356483, A356484.
%K A357279 nonn,hard,more
%O A357279 0,2
%A A357279 _Stefano Spezia_, Sep 25 2022
%E A357279 a(6) from _Michel Marcus_, May 02 2023
%E A357279 a(7)-a(10) from _Pontus von Brömssen_, Oct 14 2023