A182124 - OEIS

%I #14 Mar 19 2018 04:08:38

%S 1,1,1,7,54,958,31882,2077782,267554288,68648260400,35172685780656,

%T 36025101106326704,73784683234911510496,302228664484725680174432,

%U 2475873389968026270223227808,40564787539851948459971794384480

%N The number of simple labeled graphs on n nodes such that no two connected components have the same number of nodes.

%F E.g.f.: Product_{n>=1} (1+A001187(n)*x^n/n!) where A001187 is the number of connected labeled graphs.

%e a(4)=54 because there are 64 simple labeled graphs on 4 nodes but 10 of these have (at least) two components of the same size: * * * *; * * *-* times 6 labelings; *-* *-* times 3 labelings.

%t nn=15;g=Sum[2^Binomial[n,2]x^n/n!,{n,0,nn}];c=Range[0,nn]!CoefficientList[Series[Log[g]+1,{x,0,nn}],x];p=Product[1+c[[n+1]]x^n/n!,{n,1,nn}];Range[0,nn]!CoefficientList[Series[p,{x,0,nn}],x]

%Y Cf. A182117 (the unlabeled case).

%K nonn

%O 0,4

%A _Geoffrey Critzer_, Apr 13 2012