We study a particular number pyramid
![$ b_{n,k,l}$](abs/img1.gif)
that relates the
binomial,
Deleham, Eulerian, MacMahon-type and Stirling number triangles. The
numbers
![$ b_{n,k,l}$](abs/img1.gif)
are generated by a function
![$ B^{c}(x,y,t)$](abs/img2.gif)
,
![$ c\in \mathbb{C}$](abs/img3.gif)
, that
appears in the calculation of derivatives of a class of functions whose
derivatives can be expressed as polynomials in the function itself or a related function. Based on
the properties of the numbers
![$ b_{n,k,l}$](abs/img1.gif)
, we derive several new relations related to these triangles.
In particular, we show that the
number triangle
![$ T_{n,k}$](abs/img4.gif)
, recently constructed by Deleham (Sloane's
A088874) and
is generated by the Maclaurin series of
![$ \mathop{\rm sech}\nolimits ^{c}t$](abs/img5.gif)
,
![$ c\in \mathbb{C}$](abs/img3.gif)
.
We also give explicit expressions and various partial sums for
the triangle
![$ T_{n,k}$](abs/img4.gif)
. Further, we find that
![$ e_{2p}^{m}$](abs/img6.gif)
, the
numbers appearing in the Maclaurin series of
![$ \cosh ^{m}t$](abs/img7.gif)
, for all
![$ m\in
\mathbb{N}$](abs/img8.gif)
, equal the number of closed walks, based at a vertex, of length
![$ 2p$](abs/img9.gif)
along the edges of an
![$ m$](abs/img10.gif)
-dimensional cube.