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This can be proved by using the asymptotic growth of the central binomial coefficients, by Stirling's approximation for !, or via generating functions. The only Catalan numbers C n that are odd are those for which n = 2 k − 1; all others are even. The only prime Catalan numbers are C 2 = 2 and C 3 = 5. [1]
An example where convolutions of generating functions are useful allows us to solve for a specific closed-form function representing the ordinary generating function for the Catalan numbers, C n. In particular, this sequence has the combinatorial interpretation as being the number of ways to insert parentheses into the product x 0 · x 1 ·⋯ ...
one can generate all other Fuss–Catalan numbers if p is an integer. ... an equivalent representation expressed as a generating function can be formulated as
A slight generalization of central binomial coefficients is to take them as (+) (+) = (+,), with appropriate real numbers n, where () is the gamma function and (,) is the beta function. The powers of two that divide the central binomial coefficients are given by Gould's sequence , whose n th element is the number of odd integers in row n of ...
The method relies on two observations. First, many identities can be proved by extracting coefficients of generating functions. Second, many generating functions are convergent power series, and coefficient extraction can be done using the Cauchy residue theorem (usually this is done by integrating over a small circular contour enclosing the ...
The Catalan numbers are + (). The binomial distribution in statistics is () (). ... which is the same as the previous generating function after the substitution ...
The use of exponential generating functions (EGFs) to study the properties of Stirling numbers is a classical exercise in combinatorial mathematics and possibly the canonical example of how symbolic combinatorics is used. It also illustrates the parallels in the construction of these two types of numbers, lending support to the binomial-style ...
This is because the generating function of the Catalan numbers is not a rational function ... The new recurrence can be found by adding the generating functions for ...