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A338456 a(n) is the hafnian of a symmetric Toeplitz matrix M(2n) whose first row consists of a single zero followed by successive positive integers repeated (A004526).

%I A338456 #36 Oct 14 2023 23:56:47
%S A338456 1,1,4,45,968,34265,1799748,131572357,12770710096,1589142683313,
%T A338456 246658484353100
%N A338456 a(n) is the hafnian of a symmetric Toeplitz matrix M(2n) whose first row consists of a single zero followed by successive positive integers repeated (A004526).
%H A338456 Wikipedia, <a href="https://en.wikipedia.org/wiki/Hafnian">Hafnian</a>
%H A338456 Wikipedia, <a href="https://en.wikipedia.org/wiki/Symmetric_matrix">Symmetric matrix</a>
%H A338456 Wikipedia, <a href="http://en.wikipedia.org/wiki/Toeplitz_matrix">Toeplitz Matrix</a>
%e A338456 a(2) = 4 because the hafnian of
%e A338456 0  1  1  2
%e A338456 1  0  1  1
%e A338456 1  1  0  1
%e A338456 2  1  1  0
%e A338456 equals M_{1,2}*M_{3,4} + M_{1,3}*M_{2,4} + M_{1,4}*M_{2,3} = 4.
%t A338456 k[i_]:=Floor[i/2]; M[i_, j_, n_]:=Part[Part[ToeplitzMatrix[Array[k, 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, 5, 0]
%o A338456 (PARI) tm(n) = {my(m = matrix(n, n, i, j, if (i==1, j\2, if (j==1, i\2)))); for (i=2, n, for (j=2, n, m[i, j] = m[i-1, j-1]; ); ); m; }
%o A338456 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_, Nov 11 2020
%Y A338456 Cf. A004526.
%Y A338456 Cf. A002378 (conjectured determinant of M(2n+1)), A083392 (conjectured determinant of M(n+1)), A332566 (permanent of M(n)), A333119 (k-th super- and subdiagonal sums of the matrix M(n)).
%Y A338456 Cf. A202038, A336114, A336286, A336400.
%K A338456 nonn,hard,more
%O A338456 0,3
%A A338456 _Stefano Spezia_, Oct 28 2020
%E A338456 a(5) from _Michel Marcus_, Nov 11 2020
%E A338456 a(6)-a(10) from _Pontus von Brömssen_, Oct 14 2023