A sum of polyharmonic series 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.
 A: 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
$$F(N,k)=\sum_{j=k}^{N-1}F(j,k-1)\leq\sum_{j=k}^{N-1}P(j,k-1)
=\sum_{j=k}^{N-1}(j-(k-1))^{k-1}\leq(N-k)(N-k)^{k-1}=P(N,k).$$
A: Occasionally, by revisiting the answers to my earlier question, in which the same problem was addressed yet a computational error was made in problem posing, I found out that one can write $F(N,k)$ as
$$
\frac{1}{(N-1)!}\sum_{1\le m_1 < \dots < m_{N-1-k} < N} m_1m_2\dots m_{N-k-1}
$$
The sum here is the unsigned Stirling number of the first kind, so that all the established recurrences and estimates may apply.
I am obliged to G Cab and Servaes for helping me out with this problem. 
