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parity of #1s in binary expansion

Source: Putnam 1984 B5

September 10, 2021

number theory

Problem Statement

For each nonnegative integer

k

, let

d(k)

denote the number of

1

's in the binary expansion of

k

. Let

m

be a positive integer. Express

\sum_{k=0}^{2^m-1}(-1)^{d(k)}k^m

in the form

(-1)^ma^{f(m)}g(m)!

, where

a

is an integer and

f

and

g

are polynomials.

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