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Number of 1's in binary representation of n

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November 14, 2011
calculusfunctionnumber theorycombinatorics

Problem Statement

Let nn be a positive integer and let αn\alpha_n be the number of 11's within binary representation of nn.
Show that for all positive integers rr, 22n−αn−1∣−10∑k=−nn(2nn+k)k2r.2^{2n-\alpha_n}\phantom{-1} \bigg|^{\phantom{0}}_{\phantom{-1}} \sum_{k=-n}^{n} \binom{2n}{n+k} k^{2r}.
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