On a number pyramid related to the binomial, Deleham, Eulerian, MacMahon and Stirling number triangles

Franssens, Ghislain R.

Journal of integer sequences, Tome 9 (2006) no. 4.

Résumé

Summary: 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.

Détail

Classification : 11B83, 05A15, 11Y55, 33B10, 30B10
Keywords: generating function, binomial coefficients, deleham numbers, Eulerian numbers, macmahon numbers, Stirling numbers, Euler numbers