ClearAll[f]
f[k_, n_] := Module[{p = PadLeft[Sort /@ IntegerPartitions@k, {Automatic, k + 1}]},
Pick[p, Range[0, k].# & /@ p, n]]
Examples:
Join @@ Table[{i, #, f[i, #]} & /@ Range[i^2] /. {_, _, {}} -> Sequence[], {i, 2, 6}] //
Grid[Prepend[#, {"k", "n", "f[k,n]"}],
Dividers -> {All, Thread[{1, 2, 4, 7, 12, 19, 29, -1} -> True]},
Background -> {None, None, {{{2, 3}, {1, -1}} -> LightYellow,
{{4, 6}, {1, -1}} -> LightRed, {{7, 11}, {1, -1}} -> LightGreen,
{{12, 18}, {1, -1}} -> LightPurple, {{19, -1}, {1, -1}} -> LightBlue}}] &

{10, #, f[10, #]} & /@ Range[10^2] /. {_, _, {}} -> Sequence[] //
Grid[Prepend[#, {"k", "n", "f[k,n]"}], Dividers -> All] & // TeXForm
$$\begin{array}{|c|c|c|}
\hline
\text{k} & \text{n} & \text{f[k,n]} \\
\hline
10 & 55 & \left(
\begin{array}{ccccccccccc}
0 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\
\end{array}
\right) \\
\hline
10 & 64 & \left(
\begin{array}{ccccccccccc}
0 & 0 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 2 \\
\end{array}
\right) \\
\hline
10 & 71 & \left(
\begin{array}{ccccccccccc}
0 & 0 & 0 & 1 & 1 & 1 & 1 & 1 & 1 & 2 & 2 \\
\end{array}
\right) \\
\hline
10 & 72 & \left(
\begin{array}{ccccccccccc}
0 & 0 & 0 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 3 \\
\end{array}
\right) \\
\hline
10 & 76 & \left(
\begin{array}{ccccccccccc}
0 & 0 & 0 & 0 & 1 & 1 & 1 & 1 & 2 & 2 & 2 \\
\end{array}
\right) \\
\hline
10 & 78 & \left(
\begin{array}{ccccccccccc}
0 & 0 & 0 & 0 & 1 & 1 & 1 & 1 & 1 & 2 & 3 \\
\end{array}
\right) \\
\hline
10 & 79 & \left(
\begin{array}{ccccccccccc}
0 & 0 & 0 & 0 & 1 & 1 & 1 & 1 & 1 & 1 & 4 \\
0 & 0 & 0 & 0 & 0 & 1 & 1 & 2 & 2 & 2 & 2 \\
\end{array}
\right) \\
\hline
10 & 80 & \left(
\begin{array}{ccccccccccc}
0 & 0 & 0 & 0 & 0 & 0 & 2 & 2 & 2 & 2 & 2 \\
\end{array}
\right) \\
\hline
10 & 82 & \left(
\begin{array}{ccccccccccc}
0 & 0 & 0 & 0 & 0 & 1 & 1 & 1 & 2 & 2 & 3 \\
\end{array}
\right) \\
\hline
10 & 83 & \left(
\begin{array}{ccccccccccc}
0 & 0 & 0 & 0 & 0 & 1 & 1 & 1 & 1 & 3 & 3 \\
\end{array}
\right) \\
\hline
10 & 84 & \left(
\begin{array}{ccccccccccc}
0 & 0 & 0 & 0 & 0 & 1 & 1 & 1 & 1 & 2 & 4 \\
0 & 0 & 0 & 0 & 0 & 0 & 1 & 2 & 2 & 2 & 3 \\
\end{array}
\right) \\
\hline
10 & 85 & \left(
\begin{array}{ccccccccccc}
0 & 0 & 0 & 0 & 0 & 1 & 1 & 1 & 1 & 1 & 5 \\
\end{array}
\right) \\
\hline
10 & 86 & \left(
\begin{array}{ccccccccccc}
0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 2 & 3 & 3 \\
\end{array}
\right) \\
\hline
10 & 87 & \left(
\begin{array}{ccccccccccc}
0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 2 & 2 & 4 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 2 & 3 & 3 \\
\end{array}
\right) \\
\hline
10 & 88 & \left(
\begin{array}{ccccccccccc}
0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 1 & 3 & 4 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 2 & 2 & 4 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 3 & 3 & 3 \\
\end{array}
\right) \\
\hline
10 & 89 & \left(
\begin{array}{ccccccccccc}
0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 1 & 2 & 5 \\
\end{array}
\right) \\
\hline
10 & 90 & \left(
\begin{array}{ccccccccccc}
0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 1 & 1 & 6 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 2 & 3 & 4 \\
\end{array}
\right) \\
\hline
10 & 91 & \left(
\begin{array}{ccccccccccc}
0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 2 & 2 & 5 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 4 & 4 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 3 & 3 & 4 \\
\end{array}
\right) \\
\hline
10 & 92 & \left(
\begin{array}{ccccccccccc}
0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 3 & 5 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 4 & 4 \\
\end{array}
\right) \\
\hline
10 & 93 & \left(
\begin{array}{ccccccccccc}
0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 2 & 6 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 3 & 5 \\
\end{array}
\right) \\
\hline
10 & 94 & \left(
\begin{array}{ccccccccccc}
0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 1 & 7 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 2 & 6 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 4 & 5 \\
\end{array}
\right) \\
\hline
10 & 95 & \left(
\begin{array}{ccccccccccc}
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 3 & 6 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 5 & 5 \\
\end{array}
\right) \\
\hline
10 & 96 & \left(
\begin{array}{ccccccccccc}
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 2 & 7 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 4 & 6 \\
\end{array}
\right) \\
\hline
10 & 97 & \left(
\begin{array}{ccccccccccc}
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 8 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 3 & 7 \\
\end{array}
\right) \\
\hline
10 & 98 & \left(
\begin{array}{ccccccccccc}
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 8 \\
\end{array}
\right) \\
\hline
10 & 99 & \left(
\begin{array}{ccccccccccc}
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 9 \\
\end{array}
\right) \\
\hline
10 & 100 & \left(
\begin{array}{ccccccccccc}
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 10 \\
\end{array}
\right) \\
\hline
\end{array}$$