SimplexBasis[1] := {{1}};
SimplexBasis[N_] /; N > 1 := Module[{b = SimplexBasis[N - 1], w},
   w = Total[b]/N;
   Append[Append[#, 0] & /@ b, Append[w, Sqrt[1 - w.w]]]
   ];
ClosedSimplexBasis[N_] /; N >= 1 := Module[{
    m = SparseArray[{
       {i_, j_} /; j < i :> 1/(2 Sqrt[2 j - 1]),
       {i_, j_} /; j == i :>
        Sqrt[1 - 1/4 Sum[1/(2 k - 1), {k, 1, i - 1}]]
       }, {N, N}]
    },
   Normal[m]
   ];