Consider the following polynomial: $ (1+x+\dots+x^n)^3 $
The coefficients of the expansion for few values of $n$ ($n=1$ to $5$) are: $ 1, 3, 3, 1 $ $ 1, 3, 6, 7, 6, 3, 1 $ $ 1, 3, 6, 10, 12, 12, 10, 6, 3, 1 $ $ 1, 3, 6, 10, 15, 18, 19, 18, 15, 10, 6, 3, 1 $ $ 1, 3, 6, 10, 15, 21, 25, 27, 27, 25, 21, 15, 10, 6, 3, 1 $
Is there a closed-form formula for the $i$th element of this sequence (for different values of $n$)?
Edit This looks similar to the sequence A109439 on OEIS corresponding to the coefficients of the expansion of: $ \left( \frac{1 - x^n}{1 - x} \right)^3 .$