Seidel triangle sequences and Bi-Entringer numbers

This Seidel Triangle Sequence Calculus makes it possible to derive several three-variate generating functions, in particular for the Bi-Entringer numbers, which count the alternating permutations according to their lengths, first and last letters. The paper has been motivated by this surprising observation: the number of alternating permutations, whose last letter has a prescribed value and is greater than its first letter, is equal to the Poupard number.