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

On a Number Pyramid Related to the Binomial, Deleham, Eulerian, MacMahon and Stirling Number Triangles

Ghislain R. Franssens
Belgian Institute for Space Aeronomy
Ringlaan 3
B-1180 Brussels


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 A088874) and 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.

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(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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