The Akiyama-Tanigawa algorithm for Bernoulli numbers.
The Lucas congruence for Stirling numbers of the second kind
0. Introduction. The numbers introduced by Stirling in 1730 in his Methodus differentialis [11], subsequently called “Stirling numbers” of the first and second kind, are of the greatest utility in the calculus of finite differences, in number theory, in the summation of series, in the theory of algorithms, in the calculation of the Bernstein polynomials [9]. In this study, we demonstrate some properties of Stirling numbers of the second kind similar to those satisfied by binomial coefficients; in...
The monotone Poisson process
The coefficients of the moments of the monotone Poisson law are shown to be a type of Stirling number of the first kind; certain combinatorial identities relating to these numbers are proved and a new derivation of the Cauchy transform of this law is given. An investigation is begun into the classical Azéma-type martingale which corresponds to the compensated monotone Poisson process; it is shown to have the chaotic-representation property and its sample paths are described.
The number of topologies on a finite set.
Touchard polynomials, partial Bell polynomials and polynomials of binomial type.
Transforming recurrent sequences by using the binomial and invert operators.
Two formulas for successive derivatives and their applications.
Two very short proofs of combinatorial identity.