We study a particular number pyramid that relates the binomial, Deleham, Eulerian, MacMahon-type and Stirling number triangles. The numbers are generated by a function , , 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 , we derive several new relations related to these triangles. In particular, we show that the number triangle , recently constructed by Deleham (Sloane's A088874) and is generated by the Maclaurin series of , . We also give explicit expressions and various partial sums for the triangle . Further, we find that , the numbers appearing in the Maclaurin series of , for all , equal the number of closed walks, based at a vertex, of length along the edges of an -dimensional cube.