A369289 Number of connected graphs with loops (symmetric relations) on n unlabeled vertices with at most n edges.

1, 2, 2, 4, 8, 17, 39, 92, 227, 573, 1482, 3883, 10343, 27786, 75392, 205933, 566166, 1564316, 4342431, 12100382, 33836606, 94903889, 266914438, 752517020, 2126292931, 6020035120, 17075411671, 48514471709, 138051863755, 393397897262, 1122523343690

Offset: 0

Author

Andrew Howroyd, Feb 02 2024

Keywords

• nonn

Comments

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

Links

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

Crossrefs

Cf. A000055, A000081, A001429, A368983, A369145.

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 + 2*g(1) - 2*g(1)^2 )/2)  }

Formula

a(n) = A000055(n) + A368983(n) = A000055(n) + A000081(n) + A001429(n) for n > 0.

Inverse Euler transform of A369145.