Inspired by Robert Israel's answer: Consider the generating function of the double-indexed array $(a_{m,n})$: 
\begin{equation}
	P(s,t):=\sum_{m,n\ge0}a_{m,n}s^m t^n=\sum_{m,n\ge1}a_{m,n}s^m t^n
	=\sum_{m,n\ge1}\Big(1 + \frac{a_{m,n-1}+a_{m-1,n}}{2}\Big)s^m t^n
	=\frac{st}{(1-s)(1-t)}+\frac{s+t}2\,P(s,t), 
\end{equation}
whence
\begin{align}
	P(s,t)&=\frac{st}{(1-s)(1-t)}\frac1{1-\frac{s+t}2} \\ 
	&=\sum_{m,n\ge1}s^mt^n\sum_{k\ge0}\Big(\frac{s+t}2\Big)^k \\ 
	&=\sum_{m,n\ge1}s^mt^n\sum_{k\ge0}\Big(\frac12\Big)^k \sum_{j=0}^k\binom kj s^j t^{k-j}\\ 
	&=\sum_{p,q\ge1}s^pt^q\sum_{j=0}^{p-1}\sum_{k=j}^{q-1+j}\Big(\frac12\Big)^k \binom kj 
\end{align}
for $s,t$ in $(-1,1)$, so that 
\begin{equation}
	a_{p,q}=\sum_{j=0}^{p-1}\sum_{k=j}^{q-1+j}\Big(\frac12\Big)^k \binom kj 
\end{equation}
for natural $p,q$.