PrimitivesToFull[p_List] := Module[
{lp, h, ser},
lp = Length[p];
ser = Normal[Series[(∏_(k=1)^lp)((((1/(1-(h^k)))))^(p[[k]])),{h, 0, lp}]];
Table[Coefficient[ser, h, i], {i,0, lp}]
];
FullToPrimitives[{1}] = {};
FullToPrimitives[{1, mid___, last_}] := Module[{prev},
prev=FullToPrimitives[{1,mid}];
Append[
prev,
last - Last[PrimitivesToFull[Append[prev, 0]]]
]
]

dims={1,6,24,80,241,682,1852}

FullToPrimitives[dims]

FullToPrimitives[(2^(Range[0,7]))]

3FullToPrimitives[(2^(Range[0,6]))]+{0,0,0,1,4,12}

FullToPrimitives[(3^(Range[0,6]))]

FullToPrimitives[(4^(Range[0,6]))]