Possible Duplicate:
How to compute $\sum\limits_{k=0}^n (-1)^k{2n-k\choose k}$?
Evaluate ${2n\choose 0} - {2n-1\choose 1} + {2n-2\choose 2} - \ \cdots \ (-1)^n{n\choose n}$
The required sum is the coefficient of $x^n$ in $x^n(1-x)^{2n} + x^{n-1}(1-x)^{2n-1} \cdots (1-x)^n$
$\Rightarrow$ coef. of $x^n$ in $\frac{(1-x)^n - x(1-x)^{2n+1}}{x^2-x+1}\quad \quad \{\text{ Applying Geometric series sum formula}\}$
But then I am unable to fine the coefficient of $x^n$ in the above expression.
Please help.
Also I would appreciate any other way of solving this or any other insights.