Tuples approach

nnn = 5;
g1 = Table[
  T = Tuples[
    Table[Table[If[k == 1, 0, n/k], {n, -(k - 1), k - 1}], {k, 1,
      nn}]];
  ListLinePlot[Sort[Table[Total[T[[k]]], {k, 1, Length[T]}]]], {nn, 1,
    nnn}]


Clear[T];
nnn = 9;
m = 3;
Table[
 T = Tuples[Table[(Range[k] - 1)/k^0, {k, 1, nn}]];
 a = Sort[Table[Total[T[[k]]], {k, 1, Length[T]}]];
 Count[a, m], {nn, 0, nnn - 1}]
g1 = ListLinePlot[a];

Clear[T];
T[1, 1] = 1;
T[n_, 1] = 0;
T[n_, k_] := T[n, k] = Sum[T[n - i, k - 1], {i, 1, k - 1}];
Table[T[m + k, k], {k, 1, nnn}]
Show[g1]