differential equations - Taylor series without expanding factorial in denominator

A Taylor series is produced with the following code:

Series[Sin[x], {x, 0, 15}] 


x-x^3/6+x^5/120-x^7/5040+x^9/362880-x^11/39916800+x^13/6227020800-x^15/1307674368000+O(x^16)

Is there a simple way to prevent the factorial in each denominator from being evaluated? That is, I want an answer of the form:

$$\sin(x)=x-\frac{x^3}{3!}+\frac{x^5}{5!}+\cdots$$

Answer

s = Series[Sin[x], {x, 0, 15}] // Normal ;
s /. Times[_[a_, b_], c_] :>a (c/Inactive[Factorial][InverseFunction[Gamma][b] - 1])

Note:

The above method will not work for any function

This works for any function:

s /. Times[a_, Power[_, b_]] :> (a x^b) (b!)/Inactive[Factorial][b]

Comments

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