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Generalizing Tuenter's Binomial Sums Richard P. Brent
Mathematical Sciences Institute
Australian National University
Canberra, ACT 2614
Australia
mailto:sums@rpbrent.comsums@rpbrent.com

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Abstract:

Tuenter considered centered binomial sums of the form

$$S_r(n) = \sum_{k=0}^{2n} \binom{2n}{k}\vert n-k\vert^r,$$

where r and n are non-negative integers. We consider sums of the form

$$U_r(n) = \sum_{k=0}^n \binom{n}{k}\vert n/2-k\vert^r,$$

which are a generalization of Tuenter's sums and may be interpreted as moments of a symmetric Bernoulli random walk with n steps. The form of U_r(n) depends on the parities of both r and n. In fact, U_r(n) is the product of a polynomial (depending on the parities of r and n) times a power of two or a binomial coefficient. In all cases the polynomials can be expressed in terms of Dumont-Foata polynomials. We give recurrence relations, generating functions and explicit formulas for the functions U_r(n) and/or the associated polynomials.