Journal of Integer Sequences, Vol. 19 (2016), Article 16.3.5

Jacobsthal Decompositions of Pascal’s Triangle, Ternary Trees, and Alternating Sign Matrices

Paul Barry
School of Science
Waterford Institute of Technology


We examine Jacobsthal decompositions of Pascal's triangle and Pascal's square from a number of points of view, making use of bivariate generating functions, which we derive from a truncation of the continued fraction generating function of the Narayana number triangle. We establish links with Riordan array embedding structures. We explore determinantal links to the counting of alternating sign matrices and plane partitions and sequences related to ternary trees. Finally, we examine further relationships between bivariate generating functions, Riordan arrays, and interesting number squares and triangles.

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(Concerned with sequences A000045 A000108 A001006 A001045 A001263 A001850 A002426 A004148 A005130 A005156 A005161 A006013 A006134 A006318 A007318 A023431 A025247 A025250 A025265 A026374 A047749 A050512 A051049 A051159 A051255 A051286 A053088 A056241 A059332 A059475 A059477 A059489 A078008 A080635 A091561 A092392 A094639 A100100 A109972 A110320 A120580 A152225 A159965 A167892 A169623 A173102 A187306 A238112.)

Received July 15 2014; revised version received February 24 2016. Published in Journal of Integer Sequences, April 6 2016.

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