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Multiplicity distributions of negative binomial distribution (NBD) and modified NBD are obtained from a birth process model with an immigration. The ratio of factorial cumulant moment to factorial moment, i.e., H moment is calculated from different multiplicity distributions of NBD, modified NBD and the three fireball model, which shows an oscillatory behavior when there is a truncation of multiplicity. The oscillation of H moment is related to the common character of the truncated multiplicity distributions.