Journal of Integer Sequences, Vol. 9 (2006), Article 06.4.1 |

Belgian Institute for Space Aeronomy

Ringlaan 3

B-1180 Brussels

Belgium

**Abstract:**

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.

(Concerned with sequences A000182 A000364 A000567 A001147 A008277 A008292 A009014 A009117 A027641 A027642 A045944 A054879 A060187 A085734 A088874 and A092812 .)

Received June 17 2005;
revised version received August 8 2006.
Published in *Journal of Integer Sequences* August 21 2006.

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