A354865 - OEIS

%I #42 Oct 14 2023 23:54:08

%S 1,1,4,49,1193,50228,3098989,271913937,31382686354,4668707087571,

%T 880702869805775

%N a(n) is the hafnian of the 2n X 2n symmetric matrix whose element M_{i,j} equals phi(abs(i-j)).

%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) = M_{1,2}*M_{3,4} + M_{1,3}*M_{2,4} + M_{1,4}*M_{2,3} = 4 is the hafnian of

%e     0, 1, 1, 2;

%e     1, 0, 1, 1;

%e     1, 1, 0, 1;

%e     2, 1, 1, 0.

%t M[i_, j_, n_]:=Part[Part[Table[EulerPhi[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 (PARI) aphi(n) = n=abs(n); if(n>0, eulerphi(n), 0);

%o tm(n) = matrix(n, n, i, j, aphi(i-j));

%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. A071083 (determinant of M(n)), A085510 (permanent of M(n)).

%Y Cf. A000010, A049581, A338456.

%K nonn,hard,more

%O 0,3

%A _Stefano Spezia_, Sep 30 2022

%E a(6) from _Michel Marcus_, May 02 2023

%E a(7)-a(10) from _Pontus von Brömssen_, Oct 14 2023