%I #8 Jan 10 2021 12:42:14

%S 1,1,2,4,12,41,210,1478,17128,352926,14181309,1129005180,175491164826,

%T 52346463432414,29666505555854777,31806668884174645842,

%U 64442744342933382243031,246898165053174167804654086,1791518193851453375966274280997,24668222649527263942329934357240780

%N Number of connected graphs with vertices labeled with positive integers summing to n.

%H Andrew Howroyd, <a href="/A340025/b340025.txt">Table of n, a(n) for n = 0..50</a>

%o (PARI) \\ See A340022 for permcount, edges.

%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) = {concat([1], InvEulerT(Vec(sum(k=1, n, my(s=0); forpart(p=k, s+=permcount(p) * 2^edges(p) * x^k/prod(j=1, #p, 1 - x^p[j] + O(x^(n-k+1)))); s/k!))))}

%Y Cf. A001349, A337717, A340022, A340026, A340027.

%K nonn

%O 0,3

%A _Andrew Howroyd_, Jan 01 2021