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Could someone please help me or give me a hint on how to calculate this sum:

$$\sum_{k=0}^n \binom{n}{k}(-1)^{n-k}(x-2(k+1))^n.$$

I have been trying for a few hours now and I start thinking it may be not possible to find the answer directly, I also think it is equal to: $$x^n+(-2)^{n}n!.$$ The right answer is in fact: $$(-2)^{n}n!.$$

Thank you very much,

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As you think you have found a closed form, you could try to show it by induction. –  azimut Mar 31 '13 at 16:00
    
Are you sure it's equal? A quick calculation with anything like $n=2$ and $x=2$ for example produces different values. –  Guest 86 Mar 31 '13 at 16:05
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1 Answer 1

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Hint: The summand $(x-2(k+1))^n$ is $(-2k)^n$ plus a polynomial in $k$ of degree less than $n$. Now use the formulas here to show that your expression is $(-2)^n n!$, (without $x$).

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Thank you very much, I will try like that when I have a moment! –  Nre Mar 31 '13 at 18:51
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