Journal of Integer Sequences, Vol. 22 (2019), Article 19.3.5

Diagonal Sums in the Pascal Pyramid, II: Applications

Hacène Belbachir and Abdelghani Mehdaoui
Faculty of Mathematics
RECITS Laboratory
BP 32
El Alia 16111
Bab Ezzouar

László Szalay
University of Sopron
Institute of Mathematics
Bajcsy-Zsilinszky utca 4
H-9400 Sopron


In a previous paper, we derived a theorem which describes, in the language of linear recurrences, the sum of diagonal elements lying along a finite ray of a certain type crossing the 3-dimensional Pascal pyramid. The corresponding generating function was also determined. In this paper, we apply these results to prove several (more precisely 24) recurrence relations previously conjectured, or not given in the On-Line Encyclopedia of Integer Sequences. Moreover, we provide many (exactly 75) new summatory identities linked to sequences listed in the Encyclopedia.

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(Concerned with sequences A001850 A006134 A006442 A012000 A026375 A059304 A069835 A080609 A081671 A084605 A084768 A084769 A084770 A084771 A084772 A084773 A084774 A098269 A098270 A098332 A098333 A098334 A098336 A098337 A098338 A098339 A098340 A098341 A098409 A098410 A098411 A098430 A098439 A098443 A098444 A098453 A098455 A098456 A098479 A098480 A098658 A098659 A106186 A106258 A106259 A106260 A106261 A113179 A115864 A115865 A116091 A116092 A116093 A122868 A248168 A258723.)

Received March 30 2018; revised versions received February 4 2019; February 12 2019; March 6 2019. Published in Journal of Integer Sequences, May 19 2019.

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