Some time ago I was trying to find a closed form formula for the number of tuples $(a_k)_{k=1}^{n+s}$ of non-negative integers satisfying following conditions:

- $\sum_{k=1}^{n+s} a_k = n$,
- $\forall m \in \mathbb{N}_0 \quad m < n \implies \sum_{k=1}^{m+s} a_k > m$,

where $n \in \mathbb{N}_0 = \mathbb{N} \cup \{0\}$ and $s \in \mathbb{N}$. I think (a document I've written in Polish wasn't peer reviewed, so it'd just be my claim) I proved that the number of the mentioned tuples is given by

$$ \left( \binom{n +s}{n} \right) \frac{s}{n+s},$$

where symbol $\left( \binom{a}{b} \right)$ is a so-called multiset coefficient defined by

$$ \forall a,b \in \mathbb{N}_0 \quad \left( \binom{a}{b} \right) = \begin{cases} \binom{a+b-1}{b}, &\text{ for }a, b \in \mathbb{N}, \\ 0, &\text{ for } a = 0, b \in \mathbb{N}, \\ 1, &\text{ for } a \in \mathbb{N}_0, b = 0. \end{cases} $$

If we were to substitute $s = 1$ in the formula for the tuples, we would get the formula for the $n^\text{th}$ Catalan number.

Also, using the mentioned formula I claimed that the following holds:

$$ \forall s \in \mathbb{N} \quad \sum_{n=0}^\infty \left( 2^{-(2n+s)} \left( \binom{n +s}{n} \right) \frac{s}{n+s} \right) = 1. $$

I would like to ask whether either of these results are known (if they are correct) and if so, where could I find some paper regarding it. The proof (, which, I hope, is correct) of the first claim which I've written in the mentioned document is rather cumbersome, so I would be interested if this result could be proven in some relatively easy way.