Is there any polynomial or factorials to represent this series?

Question

I want to represent these series as a single polynomial or a form of some factorials.

$k=1$

{$1,2,2$}

$k=2$

{$1,2,3,3,3$}

$k=3$

{$1,2,3,4,4,4,4$}

$k=4$

{$1,2,3,4,5,5,5,5,5$}

Answer 1

The coefficient of $x^{j}$ in

$$[x+2x^2+\ldots+(k+1)x^{k+1}][1+(k+1)x^{k+1}(1-x)]$$

where $1 \le j \le 2k+1$.