Generalizing Tuenter’s Binomial Sums
Richard P. Brent
Mathematical Sciences Institute
Australian National University
Canberra, ACT 2614
considered centered binomial sums of the form
are non-negative integers.
We consider sums of the form
which are a generalization of Tuenter's sums and may be interpreted as
moments of a symmetric Bernoulli random walk with n
The form of Ur
) depends on the parities of both r
In fact, Ur
) is the product of a polynomial (depending on the parities
) 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 Ur
) and/or the associated polynomials.
Full version: pdf,
(Concerned with sequences
July 16 2014; revised versions received January 18 2015; January 25 2015.
Published in Journal of Integer Sequences, January 26 2015.
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