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Given a number of length $l$, the sum of the digits range from $0$ to $9l$

For each sum, there are $x$ permutations of digits. find the sequence of length $9l+1$ that solves $x$.

I found a function that solves this problem without expanding the polynomial equation $l$ times:

$f(n \in S,l,b) = \sum\limits_{i=0}^{\lfloor\frac{n}{b}\rfloor}(-1)^{i}{l \choose i}{n+l-1-bi \choose l-1}$

($l=l$, $b=10$,$S={0,1,...,9l}$)

Where $f(n,l,b)$ is the $n^{th}$ coefficient of a polynomial of length $l$ and base $b$ where $S={0,1,...,(b-1)l}$

(thanks to Tom De Medts for giving me the proper terms.)

Are there other functions that give the $n^{th}$ polynomial coefficient? I can't seem to find one.