A354865 a(n) is the hafnian of the 2n X 2n symmetric matrix whose element M_{i,j} equals phi(abs(i-j)).

1, 1, 4, 49, 1193, 50228, 3098989, 271913937, 31382686354, 4668707087571, 880702869805775

Offset

0,3

Links

Table of n, a(n) for n=0..10.

Wikipedia, Hafnian

Wikipedia, Symmetric matrix

Example

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
    0, 1, 1, 2;
    1, 0, 1, 1;
    1, 1, 0, 1;
    2, 1, 1, 0.

Mathematica

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]

Prog

(PARI) aphi(n) = n=abs(n); if(n>0, eulerphi(n), 0);
tm(n) = matrix(n, n, i, j, aphi(i-j));
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

Crossrefs

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

Cf. A000010, A049581, A338456.

Sequence in context: A121275 A329328 A188682 * A191301 A336805 A029991

Adjacent sequences: A354862 A354863 A354864 * A354866 A354867 A354868

Keyword

nonn,hard,more

Author

Stefano Spezia, Sep 30 2022

Extensions

a(6) from Michel Marcus, May 02 2023

a(7)-a(10) from Pontus von Brömssen, Oct 14 2023

Status

approved