Catalan triangle numbers and binomial coefficients

The binomial coefficients and Catalan triangle numbers appear as weight multiplicities of the finite-dimensional simple Lie algebras and affine Kac--Moody algebras. We prove that any binomial coefficient can be written as weighted sums along rows of the Catalan triangle. The coefficients in the sums form a triangular array, which we call the alternating Jacobsthal triangle. We study various subsequences of the entries of the alternating Jacobsthal triangle and show that they arise in a variety of combinatorial constructions. The generating functions of these sequences enable us to define their k-analogue of q-deformation. We show that this deformation also gives rise to interesting combinatorial sequences. The starting point of this work is certain identities in the study of Khovanov--Lauda--Rouquier algebras and fully commutative elements of a Coxeter group.