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 $B^{c}(x,y,t)$, $c\in \mathbb{C}$, 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 http://www.research.att.com/cgi-bin/access.cgi/as/ njas/sequences/eisA.cgi?Anum=A088874A088874), is generated by the Maclaurin series of $\mathop{\rm sech}\nolimits ^{c}t$, $c\in \mathbb{C}$. 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 $m\in \mathbb{N}$, equal the number of closed walks, based at a vertex, of length $2p$ along the edges of an $m$-dimensional cube.