Fix three integers $a, b, c$ and consider a sequence of integers $a_{i,j}$ defined, for $i \ge 0, j \ge 0$, recursively as follows:

$a_{i,0}=1$ for every $i$, $a_{0,j}=a+bj+cj^2$ and, for $i \ge 1, j \ge 1$, $$a_{i,j}=a_{i,j-1}+a_{i-1,j}.$$ In there a close formula for the $a_{i,j}$'s?

Fix three integers $a, b, c$ and consider a sequence of integers $a_{i,j}$ defined, for $i \ge 0, j \ge 0$, recursively as follows:

$a_{i,0}=1$ for every $i$, $a_{0,j}=a+bj+cj^2$ and, for $i \ge 1, j \ge 1$, $$a_{i,j}=a_{i,j-1}+a_{i-1,j}.$$ Is there a closed formula for the $a_{i,j}$'s?