Let $a,b\in\mathbb{N}$.
It's well known that we have $$ \int_0^v x^a (v-x)^b\,dx = \frac{a!\, b!}{(a+b+1)!} v^{a+b+1} = \frac{v^{a+b+1}}{(a+b+1){a+b \choose a}} $$
There's also closed sum expression of sorts for $\sum_{i=0}^n i^a$ (expressing it as a polynomial on $n$ of degree $a+1$ with Bernoulli numbers involved in the coefficients) known as “Faulhaber's formula”.
Is there a similar closed form expression for the following sum? $$ \sum_{i=0}^n i^a (n-i)^b $$
To be a bit clearer about what I mean by “closed form”, a sum with a number of terms dependent on $a$ and $b$ is of course acceptable, because the result is clearly a polynomial of degree $a+b+1$ in $n$, but the goal is to no longer have sums involving $n$.