Walking into an Absolute Sum

evaluates to n(£) for r = 3. In the published solution [16], It was also noted that Sx{n) = n(£), and, as a consequence, It was conjectured that S2r+i(n) equals the product of („) and a monk polynomial of degree r +1. We show this conjecture to be true, albeit with the modification of discarding the adjec-tival modifier monic. In fact, we show that S2r+l(n) = Pr(n)n(£) and S2r() = Qr()2~\ where Pr(n) and Qr(n) are both polynomials of degree r with Integer coefficients. We then Investigate the relationship of these polynomials to the Dumont-Foata polynomials [6]. These are generalizations of the Gandhi polynomials, which find their origin in a representation of the Genocchi numbers, first conjectured by Gandhi [9]. Finally, we show that the sums Sr{n) are essentially the moments of a random variate, measuring the absolute distance to the origin in a symmetric Bernoulli random walk, after In time steps.