Abstract

We study a particular number pyramid b_{n,k,l} that relates the binomial, Deleham, Eulerian, MacMahon-type and Stirling number triangles. The numbers b_{n,k,l} are generated by a function B^{c}(x, y, t), c ∈ ℂ, that appears in the calculation of derivatives of a class of functions whose derivatives can be expressed as polynomials in the function itself or a related function. Based on the properties of the numbers b_{n,k,l}, we derive several new relations related to these triangles. In particular, we show that the number triangle T_{n,k}, recently constructed by Deleham (Sloane’s \underline{A088874}), is generated by the Maclaurin series of sech^{c} t, c ∈ ℂ. We also give explicit expressions and various partial sums for the triangle T_{n,k}. Further, we find that e_{2p}^{m}, the numbers appearing in the Maclaurin series of cosh^{m} t, for all m ∈ ℕ, equal the number of closed walks, based at a vertex, of length 2p along the edges of an m-dimensional cube.