Pensieve Header: Finding Euler and inverse Euler transforms in general, and in particular, for n!.

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]]]
]
]

PrimitivesToFull[{2,1,1,1,1,1,1}]

FullToPrimitives[{1,2,4,7,12,19,30,45}]

FullToPrimitives[Table[(2^n), {n,0,100}]]

ListPlot[Log[dims=FullToPrimitives[Table[(2^n), {n,0,200}]]]]

N[((dims[[-1]])/(dims[[-2]]))]

seq=FullToPrimitives[Table[n!, {n,0,10}]]

FindSequenceFunction[seq]

Import["http://www.research.att.com/~njas/sequences/?q=1,1,4,17,92,572,4156,34159,314368,3199844&p=1&n=10&fmt=3"]