- Let $a_1(n)$ be A000108, i.e. Catalan numbers. Here $$ a_1(n)=\frac{1}{n+1}\binom{2n}{n} $$
- Let $a_2(n)$ be A059715, i.e. number of multi-directed animals on the triangular lattice. From OEIS page we know that the generating function is known in closed form. It is big and non-D-finite.
- Let $$ L_1(n, j)=\sum\limits_{p=0}^{n-j-1}\binom{j+p+1}{p+1}L_1(n-j-1, p), \\ L_1(n,n)=1 $$ Here it looks like that $L_1(n,j)$ is A033184, i.e. transposed Catalan triangle.
- Let $$ L_2(n, j)=\sum\limits_{p=0}^{n-j-1}\binom{j+p+2}{p+1}L_2(n-j-1, p), \\ L_2(n,n)=1 $$
I conjecture that $$ a_1(n)=\sum\limits_{j=0}^{n-1}L_1(n-1,j), \\ a_2(n)=\sum\limits_{j=0}^{n-1}L_2(n-1,j) $$ Is there a way to prove it? Is there a way to get closed form for $a_2(n)$ from the recursion for $L_2(n,j)$?
