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

§26.6 Other Lattice Path Numbers

… ►

Motzkin Number $M(n)$

►

M(n)

is the number of lattice paths from

(0,0)

(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)

. … ►

Table 26.6.2: Motzkin numbers

M(n)

► ►►

$n$	$M(n)$	$n$	$M(n)$	$n$	$M(n)$	$n$	$M(n)$	$n$	$M(n)$
…

► … ►

§26.6(iv) Identities

…

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.

ⓘ

Symbols:: $\in$ : element of, $c\left(\NVar{n}\right)$ : number of compositions of $n$ , $c\left(\NVar{\mathrm{condition}},\NVar{n}\right)$ : restricted number of compositions of $n$ and $T$ : set
Permalink:: http://dlmf.nist.gov/26.11.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §26.11 and Ch.26

… ►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

. …

5: 24.15 Related Sequences of Numbers

§24.15 Related Sequences of Numbers

►

§24.15(i) Genocchi Numbers

… ►

§24.15(ii) Tangent Numbers

… ►

§24.15(iii) Stirling Numbers

… ►

§24.15(iv) Fibonacci and Lucas Numbers

…

6: 26.5 Lattice Paths: Catalan Numbers

§26.5 Lattice Paths: Catalan Numbers

►

§26.5(i) Definitions

►

C\left(n\right)

is the Catalan number. … ►

§26.5(ii) Generating Function

… ►

§26.5(iii) Recurrence Relations

…

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. … ►

§26.14(iii) Identities

…

8: 26.7 Set Partitions: Bell Numbers

§26.7 Set Partitions: Bell Numbers

►

§26.7(i) Definitions

… ►

§26.7(ii) Generating Function

… ►

§26.7(iii) Recurrence Relation

… ►

§26.7(iv) Asymptotic Approximation

…

9: 26.8 Set Partitions: Stirling Numbers

§26.8 Set Partitions: Stirling Numbers

►

§26.8(i) Definitions

… ► … ►

§26.8(v) Identities

… ►

§26.8(vi) Relations to Bernoulli Numbers

…

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. …