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Note that the species module is mostly a raw translation of the old code from mupad-combinat. There are plans to update it, in particular, Jean-Baptiste Priez started to work on it, but left academia since then.

That said, why not defining H in a simpler way as follows:

sage: Z=species.SingletonSpecies()
sage: H=CombinatorialSpecies()
sage: H.define(Z+species.SetSpecies(min=2)(H))

Then, you can do, at least:

sage: g = H.generating_series()
sage: oeis([g.count(i) for i in range(10)])
0: A000311: Schroeder's fourth problem; also number of phylogenetic trees with n nodes; also number of total partitions of n.

Note that the species module is mostly a raw translation of the old code from mupad-combinat. There are plans to update it, in particular, Jean-Baptiste Priez started to work on it, but left academia since then.then. Would you be interested in working on it ?

That said, why not defining H in a simpler way as follows:

sage: Z=species.SingletonSpecies()
sage: H=CombinatorialSpecies()
sage: H.define(Z+species.SetSpecies(min=2)(H))

Then, you can do, at least:

sage: g = H.generating_series()
sage: oeis([g.count(i) for i in range(10)])
0: A000311: Schroeder's fourth problem; also number of phylogenetic trees with n nodes; also number of total partitions of n.