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A356484 a(n) is the hafnian of a symmetric Toeplitz matrix M(2*n) whose first row consists of prime(2*n), prime(2*n-1), ..., prime(1).

%I A356484 #25 Oct 14 2023 23:54:50
%S A356484 1,2,44,5210,1368900,604109562,535920536336,728155179271474,
%T A356484 1103827431509790216,2651375713654260218986,7537958658258053003685636
%N A356484 a(n) is the hafnian of a symmetric Toeplitz matrix M(2*n) whose first row consists of prime(2*n), prime(2*n-1), ..., prime(1).
%C A356484 a(n) is even for n >= 1. - _Robert Israel_, Oct 13 2023
%H A356484 Wikipedia, <a href="https://en.wikipedia.org/wiki/Hafnian">Hafnian</a>
%H A356484 Wikipedia, <a href="https://en.wikipedia.org/wiki/Symmetric_matrix">Symmetric matrix</a>
%H A356484 Wikipedia, <a href="http://en.wikipedia.org/wiki/Toeplitz_matrix">Toeplitz Matrix</a>
%e A356484 a(2) = 44 because the hafnian of
%e A356484     7  5  3  2
%e A356484     5  7  5  3
%e A356484     3  5  7  5
%e A356484     2  3  5  7
%e A356484 equals M_{1,2}*M_{3,4} + M_{1,3}*M_{2,4} + M_{1,4}*M_{2,3} = 44.
%p A356484 haf:= proc(A)
%p A356484     local n, s, Pairpart, p;
%p A356484     Pairpart := proc(L) local j, t; if L = {} then return {{}}; end if; {seq(seq({{L[1], L[j]}} union t, t = procname(L minus {L[1], L[j]})), j = 2 .. nops(L))}; end proc;
%p A356484     n := LinearAlgebra:-Dimension(A);
%p A356484     if n[1] <> n[2] then
%p A356484         error "must be square matrix";
%p A356484     end if;
%p A356484     n := n[1];
%p A356484     if n::odd then
%p A356484         error "dimension of matrix must be even";
%p A356484     end if;
%p A356484     add(mul(A[s[1], s[2]], s = p), p = Pairpart({$ (1 .. n)}));
%p A356484 end proc:
%p A356484 f:= proc(n) local i; haf(LinearAlgebra:-ToeplitzMatrix([seq(ithprime(i),i=2*n..1,-1)],symmetric)) end proc:
%p A356484 f(0):= 1:
%p A356484 map(f, [$0..7]); # _Robert Israel_, Oct 13 2023
%t A356484 k[i_]:=Prime[i]; M[i_, j_, n_]:=Part[Part[ToeplitzMatrix[Reverse[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, 6, 0]
%o A356484 (PARI) tm(n) = my(m = matrix(n, n, i, j, if (i==1, prime(n-j+1), if (j==1, prime(n-i+1))))); for (i=2, n, for (j=2, n, m[i, j] = m[i-1, j-1]; ); ); m;
%o A356484 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 A356484 Cf. A202038, A336114, A336286, A336400, A338456.
%Y A356484 Cf. A356481, A356482, A356483.
%Y A356484 Cf. A356492 (determinant of M(n)), A356493 (permanent of M(n)).
%K A356484 nonn,hard,more
%O A356484 0,2
%A A356484 _Stefano Spezia_, Aug 09 2022
%E A356484 a(6) from _Michel Marcus_, May 02 2023
%E A356484 a(7)-a(9) from _Robert Israel_, Oct 13 2023
%E A356484 a(10) from _Pontus von Brömssen_, Oct 14 2023