In this paper, a symmetric discrete distribution on integers is proposed by using a novel method of generating new distribution on Z. This distribution has only one parameter and possesses a very simple structure. Many important characteristics, such as, the recurrence relation, tail length, mode, cumulative distribution function, monotonic property of log-concavity, moments and generating functions, are investigated. Maximum likelihood estimation of the model parameter is presented. An innovative zero-inflated count regression analysis is introduced based on the proposed distribution. The utility of this model is justified by considering three applications of count data modeling and regression in comparison to some established count distributions on Z.