Diagonal sums in Pascal pyramid

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Abstract

In this paper, we describe the recurrence relation associated to the sum of diagonal elements lying along a finite ray of certain type crossing the three-dimensional Pascal pyramid. Then we determine the corresponding generating function. Finally, we deal with the so-called Morgan–Voyce phenomenon, and applying the new arguments we prove an identity, which makes it possible to prove several recurrence relations conjectured in Sloane's On-Line Encyclopedia of Integer Sequences.

Keywords

Pascal pyramid
Sum of elements
Linear recurrence
Morgan–Voyce phenomenon

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The research was supported by Hungarian National Foundation for Scientific Research Grant No. 128088.