Name of the numbers defined by $T(p,q) = T(p-1,q) + T(p,q-1)$?

Question

I came across these numbers : $$ T(p,q)= \sum_{k=0}^{q-1} {p+k-1 \choose p-1} + \sum_{l=0}^{p-1} {q+l-1 \choose q-1} \quad p,q \in \mathbb{N} $$ While trying to solve this recurrence relation : $$ \left\{ \begin{array}{c} T(p,q) = T(p-1,q) + T(p,q-1) \quad \forall p,q \in \mathbb{N}^{*} \\ T(p,0) = 1 \quad \forall p \in \mathbb{N}^{*} \\ T(0,q) = 1 \quad \forall q \in \mathbb{N}^{*} \\ \end{array} \right. $$ As both the relation and the expression are fairly simple, I'm certain these numbers have been studied already, but I can't seem to find their name, or if they have a simpler expression.

Answer 1

There is no in my knowledge a name for these numbers. Using the formula you have given, the sums can be computed and we can conclude that : $$T(p,q)={p+q-1\choose q}+{q+p-1 \choose p}={p+q \choose p}$$

(when $p$ and $q$ are both positive)