Journal of Integer Sequences, Vol. 18 (2015), Article 15.3.2 |

Mathematical Sciences Institute

Australian National University

Canberra, ACT 2614

Australia

**Abstract:**

Tuenter
considered centered binomial sums of the form

where*r* and *n* 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* 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.

where

which are a generalization of Tuenter's sums and may be interpreted as moments of a symmetric Bernoulli random walk with

(Concerned with sequences A000142 A000364 A001147 A001469 A001813 A002105 A009843 A036970 A047053 A054879 A083061 A160485 A245244.)

Received
July 16 2014; revised versions received January 18 2015; January 25 2015.
Published in *Journal of Integer Sequences*, January 26 2015.

Return to