We study a particular number pyramid bn,k,l that relates the binomial, Deleham, Eulerian, MacMahon-type and Stirling number triangles. The numbers bn,k,l are generated by a function Bc(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 Tn,k, recently constructed by Deleham (Sloane's A088874), is generated by the Maclaurin series of sechct, c ∈ ℂ. We also give explicit expressions and various partial sums for the triangle Tn,k. Further, we find that e2pm, the numbers appearing in the Maclaurin series of coshm 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.