Is there a closed form for the recurrence with $N(r,1)=N(r,r)=1$ and $N(r,c)=cN(r-1,c)+N(r-1,c-1)$ for $1<c<r$?

Question

I'm trying to find if there is a closed form solution to the recurrence relation (where $1 \leq c \leq r$):

$$ N(r,c) = \left\{ \begin{array}{lll} 1, & \text{if} & c=1,r, \\ c N(r-1,c) + N(r-1,c-1), & \text{if} & 1 < c < r, \end{array} \right. $$

This is like a deformed Pascal's triangle if you interpret $c$ as column and $r$ as row.