Observation: There exists nonnegative integers $x_1,\cdots,x_d,y_1,\cdots,y_d$ such that $a_{ij}=x_i+y_j$. Proof is left as an exercise to the reader. Now, we need to keep track of $\min\{x_1,\cdots,x_d\} + \min \{y_1,\cdots,y_d\}$ because $x_1,\cdots,x_d,y_1,\cdots,y_d$ gives the same grid as $x_1+1,\cdots,x_1+1,y_1-1,\cdots,y_d-1$ If the min is $k$ then there are $k+1$ ways to get the same grid. By stars and bars, the number of ways for the min to be at least $k$ is $\binom{n-dk+2d-1}{2d-1}$. We start with $\binom{n+2d-1}{2d-1}$ solutions. $\binom{n+d-1}{2d-1}$ have at least one duplicate, so we remove these. $\binom{n-1}{2d-1}$ has 2 duplicates, one of them counted previously, so we remove another $\binom{n-1}{2d-1}$. This gives an answer of $\binom{n+2d-1}{2d-1} - \sum\limits_{0\le j\le n+d-1, j\equiv n-1(\bmod\; d)} \binom jd$