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11: 26.5 Lattice Paths: Catalan Numbers

§26.5 Lattice Paths: Catalan Numbers

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§26.5(i) Definitions

► $C\left(n\right)$ is the Catalan number. It counts the number of lattice paths from $(0,0)$ to $(n,n)$ that stay on or above the line $y=x$. … ►

§26.5(ii) Generating Function

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12: 16.7 Relations to Other Functions

§16.7 Relations to Other Functions

… ►Further representations of special functions in terms of ${}_p{}^{}F_q^{}$ functions are given in Luke (1969a, §§6.2–6.3), and an extensive list of ${}_{q+1}{}^{}F_q^{}$ functions with rational numbers as parameters is given in Krupnikov and Kölbig (1997).

13: 23.9 Laurent and Other Power Series

… ►Let $z_0(\ne 0)$ be the nearest lattice point to the origin, and define …Explicit coefficients $c_n$ in terms of $c_2$ and $c_3$ are given up to $c_{19}$ in Abramowitz and Stegun (1964, p. 636). … ►as $t\to 0$. …Also, Abramowitz and Stegun (1964, (18.5.25)) supplies the first 22 terms in the reverted form of (23.9.2) as $1/\mathrm{\wp }\left(z\right)\to 0$. …

14: 26.6 Other Lattice Path Numbers

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Delannoy Number $D(m,n)$

► $D(m,n)$ is the number of paths from $(0,0)$ to $(m,n)$ that are composed of directed line segments of the form $(1,0)$, $(0,1)$, or $(1,1)$. … ►

Motzkin Number $M(n)$

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Narayana Number $N(n,k)$

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Schröder Number $r(n)$

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15: 24.15 Related Sequences of Numbers

§24.15 Related Sequences of Numbers

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§24.15(i) Genocchi Numbers

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§24.15(ii) Tangent Numbers

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§24.15(iii) Stirling Numbers

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§24.15(iv) Fibonacci and Lucas Numbers

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16: 27.18 Methods of Computation: Primes

§27.18 Methods of Computation: Primes

… ►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, $2^{2^n}+1$. … ►That is to say, it runs in time $O\left({(\ln n)}^c\right)$ for some constant $c$. …

17: How to Cite

How to Cite

►When citing DLMF from a formal publication, we suggest a format similar to the following: … ►

Permalinks & Reference numbers

►The direct correspondence between the reference numbers in the printed Handbook and the permalinks used online in the DLMF enables readers of either version to cite specific items and their readers to easily look them up again — in either version! … ►Note the ‘E’, ‘F’ and ‘T’ used to disambiguate equations, figures and tables. …

18: 27.12 Asymptotic Formulas: Primes

… ► $\pi \left(x\right)$ is the number of primes less than or equal to $x$. … ►

Prime Number Theorem

… ► $\pi \left(x\right)-\mathrm{li}\left(x\right)$ changes sign infinitely often as $x\to \mathrm{\infty }$; see Littlewood (1914), Bays and Hudson (2000). … ►If $a$ is relatively prime to the modulus $m$, then there are infinitely many primes congruent to $a(modm)$. … ►

19: 24.11 Asymptotic Approximations

§24.11 Asymptotic Approximations

►As $n\to \mathrm{\infty }$ ►

24.11.1 $${(-1)}^{n+1}{B}_{2n}\sim \frac{2(2n)!}{{(2\pi )}^{2n}},$$

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Symbols:

$B_n$: Bernoulli numbers, $\sim$: asymptotic equality, $\pi$: the ratio of the circumference of a circle to its diameter, $!$: factorial (as in $n!$) and $n$: integer

Referenced by:

§24.11

Permalink:

http://dlmf.nist.gov/24.11.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §24.11 and Ch.24

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24.11.2 $${(-1)}^{n+1}{B}_{2n}\sim 4\sqrt{\pi n}{\left(\frac{n}{\pi \mathrm{e}}\right)}^{2n},$$

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Symbols:

$B_n$: Bernoulli numbers, $\sim$: asymptotic equality, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm and $n$: integer

Permalink:

http://dlmf.nist.gov/24.11.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §24.11 and Ch.24

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24.11.3 $${(-1)}^n{E}_{2n}\sim \frac{2^{2n+2}(2n)!}{{\pi }^{2n+1}},$$

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Symbols:

$E_n$: Euler numbers, $\sim$: asymptotic equality, $\pi$: the ratio of the circumference of a circle to its diameter, $!$: factorial (as in $n!$) and $n$: integer

Permalink:

http://dlmf.nist.gov/24.11.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §24.11 and Ch.24

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20: 24.19 Methods of Computation

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§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 $p$

►For number-theoretic applications it is important to compute $B_{2n}(modp)$ for $2n\le p-3$; in particular to find the irregular pairs $(2n,p)$ for which $B_{2n}\equiv 0(modp)$. …