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Suppose you are given a power series $$S = \Sigma_{i=0}^{\infty}a_ix^{i}$$ with coefficients in $\mathbb{N}$, and you are tasked with telling if there can not be a finite category (or any kind of higher category) which has $a_i$ number of $i$-cells for every $i \in \mathbb{N}$, reffering to the basic counting results deduced noly from the axioms of a category (or of a certain higher category). For example: when $i \leq j$ then the "number" of $i$-cells can't be less than the "number" of $j$-cells, since any cell can be seen as an identity cell of a higher dimensions.

I suppose, at this point my question begins to go off the rail, for not too often I see a category theorist counting morphisms and above that for all Evilish matters. So, I wuold like to ask if there is some tractable version of this task, or any relevanat published results in this direction.

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Not sure if this helps, but note that the nerve of a category is 2-coskeletal, hence for $k> 2$, every $k$-sphere has a unique filler, i.e. given n k-cells, there are at most ${n\choose k+2}$ k+1-cells –  roman May 23 '13 at 6:26

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