f[depth_, factorial_] := (
   RandomChoice[{
       "n" &,
       ToString[Floor@(RandomReal[10]^2)] &,
       "(-" <> f[depth - 1, factorial] <> ")" &,
       If[factorial, "(" <> f[depth - 1, False] <> "!)",
         f[depth, False]] &,
       "(" <> f[depth - 1, factorial] <> "+" <>
         f[depth - 1, factorial] <> ")" &,
       "(" <> f[depth - 1, factorial] <> "-" <>
         f[depth - 1, factorial] <> ")" &,
       "(" <> f[depth - 1, factorial] <> "*" <>
         f[depth - 1, factorial] <> ")" &,
       "(" <> f[depth - 1, factorial] <> "/" <>
         f[depth - 1, factorial] <> ")" &,
       If[factorial,
         "(" <> f[depth - 1, factorial] <> "^" <>
          f[depth - 1, False] <> ")", f[depth, False]] &,
       If[factorial,
         "Binomial[" <> f[depth - 1, False] <> "," <>
          f[depth - 1, False] <> "]", f[depth, False]] &
       }[[1 ;; If[depth == 0, 2, -1]]]][]
   );
series = {};
While[Length@series < 100,
 func = f[4, True];
 values = Table[ToExpression@func, {n, 20}];
 values =
  values[[1 ;;
     LengthWhile[values, IntegerQ@# && # > 0 && # < 10^20 &]]];
 If[Length@values > 1 && values[[1]] != values[[2]] &&
   FreeQ[series, {_, values}],
  series = Union[series, {{func, values}}]]
 ]
series // TableForm