@jsiehler Igor Pak gave me the reference "On the complexity of computing the Tutte polynomial of bicircular matroids" (Omer Giménez and Marc Noy, Combin. Probab. Comput. 2006, https://doi.org/10.1017/S0963548305007327), which shows that counting graphs in which each component is a tree + one edge is #P-complete. Your problem is counting graphs of this type with only one component, so it doesn't directly apply, but hints that there may not be a direct formula for the answer like there is for the number of trees.