The Extended Catalan Numbers

Definition

Consider the enumeration of lattice paths in 2D plane with steps E=(1,0) and N=(0,1).
Let denote the extended Catalan numbers which equals the number of lattice paths staying weakly below .

For example, equals the number of lattice paths from (0,0) to (9,6) with the possible paths as shown in bold lines of the following figure:

Let denote the generating function for the extended Catalan numbers.

We present three equations.

The Catalan Kernel

Let denote the Catalan kernel which is the primitve root of the equation .

Using the function CatalanKernel in staircase.cpp, one can compute the Catalan kernels by

unsigned int k, l;
FormalPowerSeries<Rational<Long> > K_kl = CatalanKernel(k, l); 
std::cout << K << "\n";

Relation between the extended Catalan numbers and the Catalan kernel

.

Using the function ExtendedCatalan in staircase.cpp, one can compute the generating function for the extended Catalan numbers derived from the Catalan kernel as:

unsigned int k, l, m;
FormalPowerSeries<Rational<Long> > C_kl = ExtendedCatalan(K_kl, 1, l); 
FormalPowerSeries<Rational<Long> > C_mkml = ExtendedCatalan(K_kl, 1, m*l); 
std::cout << C_kl << "\n";

Relation between the extended Catalan numbers and the simple Catalan numbers

.

Using the function ExtendedCatalan in staircase.cpp, one can compute the generating function for the extended Catalan numbers derived from from the simple Catalan numbers as:

unsigned int k, l, m;
FormalPowerSeries<Rational<Long> > C_mCkl = ExtendedCatalan(C_kl, l, m);
std::cout << C_mCkl << "\n";