The number of connected labeled graphs with $n$ edges and $n$ nodes are listed in [OEIS A057500](https://oeis.org/A057500). 


I suspect the following must be known but I can't find a reference to it. 



> 
> **Question.** How many among the above graphs are triangle-free?
> 
> 
> 


**NOTE.** This problem is connected to [my other MO question here](https://mathoverflow.net/questions/268798/existence-of-3-distributed-subsets).



Triangle-free labeled graphs
Since a connected graph with $n$ vertices and $n$ edges is unicyclic, you just need to subtract the ones whose cycle is a triangle (<http://oeis.org/A053507>) from the total (<http://oeis.org/A057500>).


Starting at $n=4$, I get 3, 72, 1500, 32280, 748440, 18898992, 520107840, 15555704400, 503580654720, 17569154733240, ... which is not in OEIS by itself.


