Calculate:

Thanks

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- March 8th 2009, 08:24 AM #1

- March 11th 2009, 12:35 AM #2
This seems quite hard. Start by noticing that is the coefficient of in .

Therefore is the coefficient of in .

But (writing the terms in reverse order) (sum of geometric series), and this simplifies to . The coefficient of in that expression is the sum of all the binomial coefficients of powers of x in minus the sum of the coefficients in the first half of the expansion of .

But the sum of all the binomial coefficients of powers of x in is . Also, the symmetry of the binomial coefficients (the fact that ) means that the sum of the coefficients in the first half of the expansion of is .

Therefore .

- March 11th 2009, 03:28 PM #3