Count data modeling's significance and its applicability to real-world occurrences have been emphasized in a number of research studies. The purpose of this work is to introduce a new one-parameter discrete distribution for the modeling of count datasets. Some mathematical properties, such as reliability measures, characteristic function, moment-generating function, and associated measurements, such as mean, variance, skewness, kurtosis, and index of dispersion, have been derived and studied. The nature of the probability mass function and failure rate function has been studied graphically. The model parameter is estimated using renowned maximum likelihood estimation methods. A neutrosophic extension of the new model is also introduced for the modeling of interval datasets. In addition, the proposed distribution's applicability was compared to that of other discrete distributions. The study's findings show that the novel discrete distribution is a very appealing alternative to some other discrete competitive distributions.