Adding up to 36

For any integers $n \ge 1$ and $m \ge 0$ , let $S_n(m)$ the number of ordered $n$ -tuples of non-negative integers that add to $m$ . Then $S_1(m) = 1$ for all $m \ge 0$ , and $S_n(m) \; = \; \sum_{j=0}^m S_{n-1}(j) \hspace{2cm} n \ge 2\;,\;m \ge 0$ If we define the generating functions $T_n(x) \; = \; \sum_{m=0}^\infty S_n(m) x^m \hspace{2cm} |x| < 1$ then the main recurrence relation tells us that $T_n(x) \; = \; (1 - x)^{-1}T_{n-1}(x) \hspace{2cm} n \ge 2 \;,\; |x| < 1$ and hence that $T_n(x) \; = \; (1 - x)^{-n} \hspace{2cm} n \ge 1 \;, \; |x| < 1$ From this it is easy to see that $S_n(m) \; = \; \binom{n+m-1}{m} \hspace{2cm} n \ge 1\;,\; m \ge 0$ By looking at the possible values of $x_6$ , we see that the desired number is $S_5(0) + S_5(6) +S_5(12) + S_5(18) + S_5(24) + S_5(30) + S_5(36) \; = \; \boxed{167587}$