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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 formula2012-04-14
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    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.2012-04-15
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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?2012-04-15
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    The fomula is from the Laplace work. Tge formula I need is very similar, so probably, one can deduse it from Laplace' formula2012-04-16
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    What "Laplace work"? Try to meet us halfway, won't you?2012-04-16
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    Laplace, theorie analytique des probabilitied, paris, 1812, p. 1712012-04-16
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    Maybe this link http://math.stackexchange.com/q/66901/23993 will be of help.2012-04-16

3 Answers 3