# Narayana numbers

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##### 1: 24.1 Special Notation

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###### 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

… ►###### Narayana Number $N(n,k)$

► $N(n,k)$ is the number of lattice paths from $(0,0)$ to $(n,n)$ that stay on or above the line $y=x$, are composed of directed line segments of the form $(1,0)$ or $(0,1)$, and for which there are exactly $k$ occurrences at which a segment of the form $(0,1)$ is followed by a segment of the form $(1,0)$. … ► … ►###### §26.6(iv) Identities

…##### 3: 26.11 Integer Partitions: Compositions

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$c\left(n\right)$ denotes the number of compositions of $n$, and ${c}_{m}\left(n\right)$ is the number of compositions into

*exactly*$m$ parts. $c(\in T,n)$ is the number of compositions of $n$ with no 1’s, where again $T=\{2,3,4,\mathrm{\dots}\}$. … ►
26.11.1
$$c\left(0\right)=c(\in T,0)=1.$$

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►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

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►As an example, $35247816$ is an element of ${\U0001d516}_{8}.$ The

*inversion number*is the number of pairs of elements for which the larger element precedes the smaller: … ► ►The*Eulerian number*, denoted $$, is the number of permutations in ${\U0001d516}_{n}$ with exactly $k$ descents. …The Eulerian number $$ is equal to the number of permutations in ${\U0001d516}_{n}$ 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

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