smc

Note that the first differences create a linear function, so the sequence $u_n$ is quadratic. The first three terms of the sequence are $5$, $8$, and $15$. From there, a system of equations can be written. $$a+b+c=5$$ $$4a+2b+c=8$$ $$9a+3b+c=15$$ Solve the system to get $a=2$, $b=-3$, and $c=6$. The sum of the coefficients is $\boxed{\text{5}}$.