A340026 Number of connected graphs with n integer labeled vertices covering an initial interval of positive integers.

1, 1, 2, 12, 151, 3845, 192215, 18642053, 3534415032, 1322914720382, 983866402492022, 1458669558830420947, 4317992152324160500565, 25541957673530923214876165, 302031658361424323818453728818, 7141206379474081326199747144178588, 337646560987347470614138636684815527025

Offset: 0

Author

Andrew Howroyd, Jan 02 2021

Keywords

• nonn

Examples

			a(2) = 2 because there is 1 connected graph on 2 vertices which can either have both vertices labeled 1 or one vertex labeled 1 and the other 2.
a(3) = 12 because there are 2 connected graphs on 3 vertices. The complete graph K_3 can be labeled in 4 ways (111, 112, 122, 123) and the path graph P_3 can be labeled in 8 ways (111, 112, 121, 122, 212, 123, 132, 213).
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 0..50

Crossrefs

Cf. A340023, A340025, A340027.

Programs

PARI

\\ See A340023 for G(n,k).
InvEulerT(v)={my(p=log(1+x*Ser(v))); dirdiv(vector(#v,n,polcoef(p,n)), vector(#v,n,1/n))}
seq(n)={my(v=concat([1], InvEulerT(vector(n, n, G(n, y))))); sum(k=0, n, subst(v, y, k)*sum(r=k, n, binomial(r, k)*(-1)^(r-k)))}