Pensieve header: The symmetric algebra over a symmetric algebra has exponential growth.

(* Primitives To All *)
P2A[p_List] := Module[{x},
CoefficientList[Series[
Product[(1-x^i)^(-p[[i]]), {i, Length[p]}],
{x, 0, Length[p]}],x]
]

P2A[{5,0,0,0,0,0,0,0,0}]

t1=P2A[{5}~Join~Table[0,{1000}]]

Rest[t1] // P2A // Log // ListPlot