OEIS A000344: Fivefold Catalan Convolution, Lattice Paths, and Young Tableaux

A000344 counts a surprising blend of combinatorics and algebra. It arises as the number of lattice paths from (0,0) to (n,n) that touch but never cross the line x - y = 2 (i.e., stay on or below x - y = 2), which is the 5-fold convolution of the Catalan numbers. Equivalently, it tallies standard Young tableaux of shape (n+2, n, 2), and its ordinary generating function is A(z) = z^2 C(z)^5, where C(z) is the Catalan generating function. We’ll sketch the combinatorial picture, connect to the 5-fold Catalan convolution, mention the d-finite (finite-recurrence) structure that helps with computation, and discuss the asymptotic growth ~ const · 4^n / n^{7/2}. Finally, we’ll pose the natural question: what happens if you shift the boundary even further? Note:  This podcast was AI-generated, and sometimes AI can make mistakes.  Please double-check any critical information.Sponsored by Embersilk LLC