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.