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Show that following formula is true:

$$ \sum_{i=0}^{[n/2]}(-1)^i (n-2i)^n{n \choose i}=2^{n-1}n! $$

Using formula calculate $$ \sum_{i=0}^n(2i-n)^p{p \choose i} $$

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for n = 2 i get 0 = 2 from the first formula so it seems ill-formed. Please check the formula –  Hardy Apr 14 '12 at 23:53
    
So, why do you think the formula is true? Why do you think you can use it to calculate the second sum? Did you read this somewhere? If so, you could tell us where, it might help us get some traction. –  Gerry Myerson Apr 15 '12 at 9:20
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This is very similar to your last question. What is the source for this? Can you tell us what you've tried? –  Antonio Vargas Apr 15 '12 at 23:17
    
The fomula is from the Laplace work. Tge formula I need is very similar, so probably, one can deduse it from Laplace' formula –  Michael Apr 16 '12 at 2:46
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Maybe this link math.stackexchange.com/q/66901/23993 will be of help. –  David Apr 16 '12 at 5:11
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