Force Mathematica to display `Series` in factorial notation

You can do this with some tinkering with Inactive, since that's one of the ways to prevent the factorial from evaluating:

factorialForm[s : HoldPattern @ SeriesData[x_, x0_, coeffList_List, n0_, n1_, 1]] :=
     With[{powRange = Range[n0, n0 + Length[coeffList] - 1]},
      SeriesData[x, x0,
       Divide[
        coeffList * Factorial[powRange],
        Map[Inactive[Factorial], powRange]
       ],
       n0, n1, 1
      ]
     ];
factorialForm[Series[Sin[x], {x, 0, 10}]]

Instead of Inactive[Factorial] there are other holding constructs you can use, like Defer[Factorial[#]]& or HoldForm[Factorial[#]] &. The nice thing about Inactive is that you can easily get rid of it again with Activate.

I only implemented this for SeriesData[__, 1]. You'd have to do a little thinking if you to make it work for other step sizes.

Also possible is:

$Post = #1 /. HoldPattern[SeriesData[Verbatim[z__]]] :> SeriesData[z] /. 
 Rational[a_, b_] :> 
   With[{invf = Reduce`FactorialInverse[b][[1]]}, 
    a/HoldForm[invf!]] & ; 

Then

Series[Sin[Pi*x],{x,0,10}]

displays as desired.

Here is a function you can use to replace Series[]:

inactiveSeries[f_, {x_, x0_, n_}] := Module[{kk, tc},
        tc[kk_] = Inactivate[Evaluate[SeriesCoefficient[f, {x, x0, kk}]], 
                             Factorial | Gamma | Pochhammer];
        Sum[tc[kk] (x - x0)^kk, {kk, 0, n}] + O[x, x0]^(n + 1)]

For example,

inactiveSeries[Sin[π x], {x, 0, 10}]

A more complicated example:

inactiveSeries[Hypergeometric1F1Regularized[-1/3, 1/5, x], {x, 0, 3}]