Given $n \in \mathbb{N}$, I am interested in the vectors $(v_1, \dots, v_m)$ with $v_i \in \mathbb{N}$ for $1\leq i \leq m$, such as the linear diophantine equation:
$$ \sum_{i=1}^{m} iv_i = n $$ holds.
For instance there are 7 vectors such that for n=5, and avoiding zeros at the right:
(5)
(3,$\,$1)
(1,2)
(2,0,1)
(0,1,1)
(1,0,0,1)
(0,0,0,0,1)
Is there a explicit formula for determining the number of vectors with this property for a given n? I have found a recursive formula but it would be great is the explicit versions exists.
The recursive definition says that the number of vectors that represent n is s(n,n), where $$ s(n,m) = \begin{cases} 0 & \text{if $n=0$} \\ 1 & \text{if $n>0, m=1$} \\ 1+ \sum_{i=0}^{i\leq \frac{n}{m}} s(n-i\cdot m, m-1) & \text{if $n>0, m>1, n \mod m=0$} \\ \sum_{i=0}^{i\leq \frac{n}{m}} s(n-i\cdot m, m-1) & \text{if $n>0, m>1, n \mod m \neq 0$} \end{cases} $$ For instance $s(5,5)=7$ because $$s(5,5) = 1 + \underbrace{s(5,4)}_{6} + \underbrace{s(0,4)}_{0}$$ I would also like to know if the vectors with this property are well-known in the sense that they have been studied before. Thanks!