A368983 Number of connected graphs with loops (symmetric relations) on n unlabeled vertices with n edges.

1, 1, 1, 3, 6, 14, 33, 81, 204, 526, 1376, 3648, 9792, 26485, 72233, 198192, 546846, 1515687, 4218564, 11782427, 33013541, 92759384, 261290682, 737688946, 2086993034, 5915398230, 16795618221, 47763406249, 136028420723, 387928330677, 1107692471888, 3166613486137

Offset: 0

Author

Andrew Howroyd, Jan 11 2024

Keywords

• nonn

Comments

The graphs considered here can have loops but not parallel edges.

Examples

			Representatives of the a(3) = 3 graphs are:
   {{1,2}, {1,3}, {2,3}},
   {{1}, {1,2}, {1,3}},
   {{1}, {1,2}, {2,3}}.
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 0..500

Crossrefs

A diagonal of A322114.

The labeled version is A368951.

Cf. A000081, A001429, A027852, A068051, A283755, A368984 (not necessarily connected).

Programs

PARI

\\ TreeGf gives gf of A000081.
TreeGf(N)={my(A=vector(N, j, 1)); for (n=1, N-1, A[n+1] = 1/n * sum(k=1, n, sumdiv(k, d, d*A[d]) * A[n-k+1] ) ); x*Ser(A)}
seq(n)={my(t=TreeGf(n)); my(g(e)=subst(t + O(x*x^(n\e)), x, x^e) + O(x*x^n)); Vec(1 + (sum(d=1, n, eulerphi(d)/d*log(1/(1-g(d)))) + ((1+g(1))^2/(1-g(2))-1)/2 - (g(1)^2 + g(2)))/2)}

Formula

a(n) = A000081(n) + A001429(n) = A068051(n) - A027852(n) for n > 0.