Asymptotic normality of the Stirling-Whitney-Riordan triangle

Abstract

Recently, Zhu [34] introduced a Stirling-Whitney-Riordan triangle [Tn,k]n,k?0 satisfying the recurrence Tn,k = (b1k + b2)Tn?1,k?1 + [(2?b1 + a1)k + a2 + ?(b1 + b2)]Tn?1,k + ?(a1 + ?b1)(k + 1)Tn?1,k+1, where initial conditions Tn,k = 0 unless 0 ? k ? n and T0,0 = 1. Denote by Tn = Pnk =0 Tn,k. In this paper, we show the asymptotic normality of Tn,k and give an asymptotic formula of Tn. As applications, we show the asymptotic normality of many famous combinatorial numbers, such as the Stirling numbers of the second kind, the Whitney numbers of the second kind, the r-Stirling numbers and the r-Whitney numbers of the second kind.