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
Examples
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.
Links
- Wikipedia, Hafnian
- Wikipedia, Symmetric matrix
Crossrefs
Programs
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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] -
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
Extensions
a(6) from Michel Marcus, May 02 2023
a(7)-a(10) from Pontus von Brömssen, Oct 14 2023