I have $(s+s^2+s^3+s^4+s^5+s^6)^7$, and I'm trying to find the coefficient on $s^{14}$. I've tried using the multinomial theorem, but that leads to the problem of finding all $k_1, k_2, \ldots , k_6$ such that $\sum_{n=1}^6 k_n = 7$ and $\sum_{n=1}^6nk_n = 14$, and that doesn't seem to put me any closer to an answer.
I tried rewriting it as $s^7(1+s+s^2+s^3+s^4+s^5)^7$ and looking for the coefficient of $s^7$ in the right half, $(1+s+s^2+s^3+s^4+s^5)^7$, but that doesn't make things much easier.
This is equivalent to counting the number of ways of writing $14$ as an ordered sum of $7$ summands from $1$ to $6$, which is equal to the number of ways of distributing $14$ balls over $7$ non-empty bins with capacity $6$, which is equal to the number of ways of distributing $14-7=7$ balls over $7$ bins with capacity $5$, which is given by
$$ \binom{7+7-1}{7-1}-7\binom{7+7-1-6}{7-1}=\binom{13}6-7\cdot7=1716-49=1667\;, $$
where the second term subtracts the configurations counted by the first term that exceed one of the capacity restrictions (see also Balls In Bins With Limited Capacity).