FullToPrimitives from Pensieve/2009-12:

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

FullToPrimitives[Table[2^k,{k,0,10}]]

Import["http://oeis.org/search?q=2,1,2,3,6,9,18,30,56,99"]

FullToPrimitives[Table[3^k,{k,0,10}]]

Import["http://oeis.org/search?q=3,3,8,18,48,116,312,810,2184,5880"]