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%I #7 Jan 10 2021 12:42:27
%S 1,1,2,12,151,3845,192215,18642053,3534415032,1322914720382,
%T 983866402492022,1458669558830420947,4317992152324160500565,
%U 25541957673530923214876165,302031658361424323818453728818,7141206379474081326199747144178588,337646560987347470614138636684815527025
%N Number of connected graphs with n integer labeled vertices covering an initial interval of positive integers.
%H Andrew Howroyd, <a href="/A340026/b340026.txt">Table of n, a(n) for n = 0..50</a>
%e 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.
%e 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).
%o (PARI) \\ See A340023 for G(n,k).
%o InvEulerT(v)={my(p=log(1+x*Ser(v))); dirdiv(vector(#v,n,polcoef(p,n)), vector(#v,n,1/n))}
%o 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)))}
%Y Cf. A340023, A340025, A340027.
%K nonn
%O 0,3
%A _Andrew Howroyd_, Jan 02 2021
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