Could anyone suggest how to obtain good estimates from above $\vee$ below for the following finite series: $$F(N,k):=\sum_{n_1 + \dots + n_k < N, \; n_i\ge 1}\frac{1}{n_1(n_1+n_2)\dots(n_1 + \dots + n_k)}?$$ For $k=1$ this is just a harmonic series. One may sort of approximate it by an integral $$ \int_{1}^{N-k}x_1^{-1}\int_{x_1+1}^{N-k+1}x_2^{-1}\dots \int_{x_{k-1}+1}^{N-1}x_k\, dx_k\dots dx_1, $$ but this yields neither an upper nor a lower bound that easily.
Alternatively, I am looking for estimates for expanded expressions $\Phi(N,k) = \sum_{j=0}^{k-1}F(N,j)$ with $F(N,0)=1$, or even $\Theta(N,k) = \sum_{i=0}^{L}\frac{\Phi(ik,k)}{ik}$, as some neat cancellations perhaps happen there.
Not a complete answer, some relatively simple bounds on $F(N,k)$. The sum $F(N,k)$ runs over all increasing sequences of length $k$ in $\{1,\ldots,N-1\}$. There are precisely $\binom{N-1}{k}$ such sequences, and clearly each summand is at most $\tfrac{1}{k!}$, which shows that $$F(N,k)\leq\frac{1}{k!}\binom{N-1}{k}.$$ Similarly, a lower bound $F(N,k)$ comes from the fact that every summand is bounded below by $$\frac{1}{N(N-1)(N-2)\cdots(N-k+1)}=\frac{(N-k)!}{N!},$$ which shows that $$F(N,k)\geq\frac{(N-k)!}{N!}\binom{N-1}{k}=\frac{N-k}{N(k!)}=\frac{1}{k!}-\frac{1}{(k-1)!N}$$
Alternatively, here's a proof by induction on $k$ that $F(N,k)\leq(N-k)^k$ for all $N,k\geq1$: For $k=1$ and all $N\geq1$ we have $$F(N,1)=H_{N-1}\leq(N-1)^1= P(N,1).$$ If $F(j,k-1)\leq P(j,k-1)$ for all $j\in\Bbb{N}$ then $$F(N,k)=\sum_{j=k}^{N-1}F(j,k-1)\leq\sum_{j=k}^{N-1}P(j,k-1) =\sum_{j=k}^{N-1}(j-(k-1))^{k-1}\leq(N-k)(N-k)^{k-1}=P(N,k).$$