It is useful to think about (finitary) monads on sets as algebraic theories. An algebraic theory consists of a set of function symbols together with their arities and a set of axioms of the form $t = s$, where $t$ and $s$ are terms constructed from variables and function symbols. For every such theory, we can define a monad $T(X)$ as the set of terms of this theory with variables in $X$ up to equivalence generated by the axioms. If for every finite set $X$ the set $T(X)$ is also finite, then this gives us a monad on finite sets and every monad can be presented in this form.
Now, there are two exceptional theories:
- The theory with no function symbols and the only axiom $x = y$. This theory generates sequence $0,1,1,1,1,\ldots$.
- The theory with a single constant $c$ and the only axiom $x = y$. This theory generates sequence $1,1,1,1,1,\ldots$.
Every other theory must generate an increasing sequence. Indeed, every term with $n$ variables $x_1, \ldots x_n$ is also a term with $n+1$ variables $x_1, \ldots x_{n+1}$ in which $x_{n+1}$ is not used. Moreover, we have the term $x_{n+1}$, so if there are $a_n$ terms with $n$ variables, then there is at least $a_n+1$ terms with $n+1$ variables.
Of course, there are more restrictions. For example, $a_2 \geq 2a_1-a_0$. Indeed, for every term $t(x)$, we have terms $x,y \mapsto t(x)$ and $x,y \mapsto t(y)$ which can be equal only when $t(x)$ is constant. Similarly, there are lower bound on every $a_n$. I think that it should be true that
$$ a_n \geq \binom{n}{n-1}a_{n-1} - \binom{n}{n-2}a_{n-2} + \ldots + (-1)^{n+1} \binom{n}{0} a_0 $$
for every $n > 1$ (and for $n = 1$, the lower bound is $a_0+1$), but I did not check this carefully.
Finally, I think these lower bound are the only restriction for sequences with $a_0 > 0$. If we have a theory with a constant $c$, we can always add exactly $k$ terms with $n$ variables to it (for every $n$ and $k$) without modifying $a_i$ for $i < n$. We just add $k$ function symbols $f_1, \ldots f_k$ of arity $n$ and axioms of the form $f_i(x_1, \ldots x_n) = f_i(x_{\sigma 1}, \ldots x_{\sigma n})$ for every $i$ and every permutation $\sigma$ and $t = c$ for every term $t$ such that its top function symbol is $f_i$ for some $i$ and either $t$ has at least two function symbols or at least one variable occurs twice in $t$.