# Motzkin numbers

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##### 1: 24.1 Special Notation
###### Bernoulli Numbers and Polynomials
The origin of the notation $B_{n}$, $B_{n}\left(x\right)$, is not clear. …
###### Euler Numbers and Polynomials
Its coefficients were first studied in Euler (1755); they were called Euler numbers by Raabe in 1851. The notations $E_{n}$, $E_{n}\left(x\right)$, as defined in §24.2(ii), were used in Lucas (1891) and Nörlund (1924). …
##### 2: 26.6 Other Lattice Path Numbers
###### MotzkinNumber$M(n)$
$M(n)$ is the number of lattice paths from $(0,0)$ to $(n,n)$ that stay on or above the line $y=x$ and are composed of directed line segments of the form $(2,0)$, $(0,2)$, or $(1,1)$. …
##### 3: 26.11 Integer Partitions: Compositions
$c\left(n\right)$ denotes the number of compositions of $n$, and $c_{m}\left(n\right)$ is the number of compositions into exactly $m$ parts. $c\left(\in\!T,n\right)$ is the number of compositions of $n$ with no 1’s, where again $T=\{2,3,4,\ldots\}$. …
26.11.1 $c\left(0\right)=c\left(\in\!T,0\right)=1.$
The Fibonacci numbers are determined recursively by … Additional information on Fibonacci numbers can be found in Rosen et al. (2000, pp. 140–145).
##### 4: 27.18 Methods of Computation: Primes
###### §27.18 Methods of Computation: Primes
An overview of methods for precise counting of the number of primes not exceeding an arbitrary integer $x$ is given in Crandall and Pomerance (2005, §3.7). …An analytic approach using a contour integral of the Riemann zeta function (§25.2(i)) is discussed in Borwein et al. (2000). … These algorithms are used for testing primality of Mersenne numbers, $2^{n}-1$, and Fermat numbers, $2^{2^{n}}+1$. …
##### 6: 26.5 Lattice Paths: Catalan Numbers
###### §26.5(i) Definitions
$C\left(n\right)$ is the Catalan number. …
##### 7: 26.14 Permutations: Order Notation
As an example, $35247816$ is an element of $\mathfrak{S}_{8}.$ The inversion number is the number of pairs of elements for which the larger element precedes the smaller: … The Eulerian number, denoted $\genfrac{<}{>}{0.0pt}{}{n}{k}$, is the number of permutations in $\mathfrak{S}_{n}$ with exactly $k$ descents. …The Eulerian number $\genfrac{<}{>}{0.0pt}{}{n}{k}$ is equal to the number of permutations in $\mathfrak{S}_{n}$ with exactly $k$ excedances. …
##### 10: 24.19 Methods of Computation
###### §24.19(i) Bernoulli and Euler Numbers and Polynomials
Equations (24.5.3) and (24.5.4) enable $B_{n}$ and $E_{n}$ to be computed by recurrence. …A similar method can be used for the Euler numbers based on (4.19.5). …
###### §24.19(ii) Values of $B_{n}$ Modulo $p$
We list here three methods, arranged in increasing order of efficiency. …