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\begin{document}

\begin{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
\seqnum{A088874}),
is generated by the Maclaurin series of $\func{sech}^{c}t$,
$ 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.
\end{abstract}

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