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Find a sum

Source: Romania TST 5 2012, Problem 2

May 17, 2012
geometric seriesalgebra proposedalgebra

Problem Statement

Let nn be a positive integer. Find the value of the following sum ∑(n)∑k=1nek2e1+⋯+ek−2k−n,\sum_{(n)}\sum_{k=1}^n {e_k2^{e_1+\cdots+e_k-2k-n}}, where ek∈{0,1}e_k\in\{0,1\} for 1≤k≤n1\leq k \leq n, and the sum ∑(n)\sum_{(n)} is taken over all 2n2^n possible choices of e1,…,ene_1,\ldots ,e_n.
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