Many famous families of integers can be represented by the number of paths through a lattice given various restrictions. This Demonstration illustrates the various paths from to using northeast and southeast steps, with the option to permit no eastward steps, single steps east, or double steps east, as well as the option to accept paths dipping below the axis (southeast steps). Counting the paths for the various choices generates six named sequences.
Snapshot 1: paths consisting only of steps northeast and southeast that are allowed to drop below the axis represent the central binomial coefficients—that is, the middle entries in Pascal's triangle
Snapshot 2: paths that do not cross the axis and consist of single steps northeast and southeast and double steps east describe the Schröder numbers
Snapshot 3: paths that can cross the axis and consist only of single steps northeast, southeast, and east are equivalent to the sums of squares of the trinomial coefficients—that is, the coefficients in the expansion of
Identification of "sums of squares of trinomial coefficients" (A082758) thanks to the On-Line Encyclopedia of Integer Sequences: oeis.org.