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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]
Whilst the above is a concrete example Catalan numbers, similar problems can be evaluated using Fuss-Catalan formula: Computer Stack: ways of arranging and completing a computer stack of instructions, each time step 1 instruction is processed and p new instructions arrive randomly. If at the beginning of the sequence there are r instructions ...
In mathematics, a generating function is a representation of an infinite sequence of numbers as the coefficients of a formal power series.Generating functions are often expressed in closed form (rather than as a series), by some expression involving operations on the formal series.
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 Catalan numbers are ... this remains true if n is any number and k is such that all the numbers between 1 and k are ... the ordinary generating function of the ...
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 ...
An infinite series of any rational function of can be reduced to a finite series of polygamma functions, by use of partial fraction decomposition, [8] as explained here. This fact can also be applied to finite series of rational functions, allowing the result to be computed in constant time even when the series contains a large number of terms.
In the typical special case that R = 2, this sequence coincides with the Catalan numbers 1, 2, 5, 14, etc. In particular, the second moment is R 2 ⁄ 4 and the fourth moment is R 4 ⁄ 8, which shows that the excess kurtosis is −1. [1] As can be calculated using the residue theorem, the Stieltjes transform of the Wigner distribution is given by