I'm very new to Mathematica, so excuse my innocence. I have the following expression:

$$ \left( \sum_{n=0}^r \frac{(-1)^n}{n!} y^n \right)^f $$

I would like Mathematica to expand out the expression in powers of $y$, as a polynomial, where $r$ and $f$ are arbitrary positive integers. Is this possible?

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    – user9660
    Commented May 22, 2016 at 7:36
  • $\begingroup$ People here generally like users to post code as Mathematica code instead of images or TeX, so they can copy-paste it. It makes it convenient for them and more likely you will get someone to help you. $\endgroup$
    – user9660
    Commented May 22, 2016 at 7:37
  • $\begingroup$ Take the result from a given r, f (here called result), then Series[result, {y, Infinity, 0}] // Normal $\endgroup$
    – ciao
    Commented May 22, 2016 at 7:41
  • 1
    $\begingroup$ That will give an expression in terms of specific $r$,$f$. I desire an expression for general $r$ and $f$...In other words ungiven. $\endgroup$ Commented May 22, 2016 at 8:10

1 Answer 1


A bit complicated, this one:

With[{m = 5, r = 3}, 
     CoefficientList[Sum[(-x)^n/n!, {n, 0, m}]^r, x] == 
     Table[Sum[FactorialPower[r, k]
               BellY[n, k, Table[(-1)^i, {i, m}]],
               {k, 0, r}]/n!, {n, 0, m r}]]

Recall that the partial Bell polynomials are a way to express Faà di Bruno's formula, which applies here since the coefficients of a polynomial are the same as the successive derivatives of a polynomial divided by an appropriate factorial. There may be a less cumbersome closed form; I'll keep trying to look.

  • $\begingroup$ Thanks a lot for that! A very good idea, I will examine it more closely. $\endgroup$ Commented May 23, 2016 at 4:03

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