Narayana numbers

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21: 24.6 Explicit Formulas

§24.6 Explicit Formulas

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24.6.1 $$B_{2n}=\sum _{k=2}^{2n+1}\frac{{(-1)}^{k-1}}{k}\left(\genfrac{}{}{0pt}{}{2n+1}{k}\right)\sum _{j=1}^{k-1}{j}^{2n},$$

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Symbols:

$B_n$: Bernoulli numbers, $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, $j$: integer, $k$: integer and $n$: integer

Referenced by:

§24.6

Permalink:

http://dlmf.nist.gov/24.6.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §24.6 and Ch.24

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24.6.4 $$E_{2n}=\sum _{k=1}^n\frac{1}{2^{k-1}}\sum _{j=1}^k{(-1)}^j\left(\genfrac{}{}{0pt}{}{2k}{k-j}\right)j^{2n},$$

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Symbols:

$E_n$: Euler numbers, $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, $j$: integer, $k$: integer and $n$: integer

Referenced by:

§24.6

Permalink:

http://dlmf.nist.gov/24.6.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §24.6 and Ch.24

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24.6.9 $$B_n=\sum _{k=0}^n\frac{1}{k+1}\sum _{j=0}^k{(-1)}^j\left(\genfrac{}{}{0pt}{}{k}{j}\right)j^n,$$

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Symbols:

$B_n$: Bernoulli numbers, $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, $j$: integer, $k$: integer and $n$: integer

Referenced by:

§24.6

Permalink:

http://dlmf.nist.gov/24.6.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §24.6 and Ch.24

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24.6.12 $$E_{2n}=\sum _{k=0}^{2n}\frac{1}{2^k}\sum _{j=0}^k{(-1)}^j\left(\genfrac{}{}{0pt}{}{k}{j}\right){(1+2j)}^{2n}.$$

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Symbols:

$E_n$: Euler numbers, $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, $j$: integer, $k$: integer and $n$: integer

Referenced by:

§24.6, §24.6

Permalink:

http://dlmf.nist.gov/24.6.E12

Encodings:

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See also:

Annotations for §24.6 and Ch.24

22: 27.13 Functions

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§27.13(i) Introduction

►Whereas multiplicative number theory is concerned with functions arising from prime factorization, additive number theory treats functions related to addition of integers. …The subsections that follow describe problems from additive number theory. … ►

§27.13(ii) Goldbach Conjecture

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§27.13(iii) Waring’s Problem

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23: 26.13 Permutations: Cycle Notation

… ►The Stirling cycle numbers of the first kind, denoted by $\left[\genfrac{}{}{0.0pt}{}{n}{k}\right]$, count the number of permutations of $\{1,2,\mathrm{\dots },n\}$ with exactly $k$ cycles. They are related to Stirling numbers of the first kind by …See §26.8 for generating functions, recurrence relations, identities, and asymptotic approximations. … ►The derangement number, $d(n)$, is the number of elements of ${𝔖}_n$ with no fixed points: … ►A permutation is even or odd according to the parity of the number of transpositions. …

24: 27.9 Quadratic Characters

§27.9 Quadratic Characters

►For an odd prime $p$, the Legendre symbol $(n|p)$ is defined as follows. … ►

27.9.2 $$(2|p)={(-1)}^{(p^2-1)/8}.$$

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Symbols:

$(n|p)$: Legendre symbol and $p$: odd prime

Referenced by:

§27.9

Permalink:

http://dlmf.nist.gov/27.9.E2

Encodings:

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See also:

Annotations for §27.9 and Ch.27

►If $p,q$ are distinct odd primes, then the quadratic reciprocity law states that … ►If an odd integer $P$ has prime factorization $P={\prod }_{r=1}^{\nu \left(n\right)}{p}_r^{a_r}$, then the Jacobi symbol $(n|P)$ is defined by $(n|P)={\prod }_{r=1}^{\nu \left(n\right)}{(n|p_r)}^{a_r}$, with $(n|1)=1$. …

25: 24.21 Software

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§24.21(ii) $B_n$, $B_n\left(x\right)$, $E_n$, and $E_n\left(x\right)$

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26: 27.19 Methods of Computation: Factorization

… ►Techniques for factorization of integers fall into three general classes: Deterministic algorithms, Type I probabilistic algorithms whose expected running time depends on the size of the smallest prime factor, and Type II probabilistic algorithms whose expected running time depends on the size of the number to be factored. … ►As of January 2009 the largest prime factors found by these methods are a 19-digit prime for Brent–Pollard rho, a 58-digit prime for Pollard $p-1$, and a 67-digit prime for ecm. … ►These algorithms include the Continued Fraction Algorithm (cfrac), the Multiple Polynomial Quadratic Sieve (mpqs), the General Number Field Sieve (gnfs), and the Special Number Field Sieve (snfs). …The snfs can be applied only to numbers that are very close to a power of a very small base. The largest composite numbers that have been factored by other Type II probabilistic algorithms are a 63-digit integer by cfrac, a 135-digit integer by mpqs, and a 182-digit integer by gnfs. …

27: 26.17 The Twelvefold Way

… ►The twelvefold way gives the number of mappings $f$ from set $N$ of $n$ objects to set $K$ of $k$ objects (putting balls from set $N$ into boxes in set $K$). …In this table ${\left(k\right)}_n$ is Pochhammer’s symbol, and $S(n,k)$ and $p_k\left(n\right)$ are defined in §§26.8(i) and 26.9(i). … ►

Table 26.17.1: The twelvefold way.

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elements of $N$	elements of $K$	$f$ unrestricted	$f$ one-to-one	$f$ onto
labeled	labeled	$k^n$	${\left(k-n+1\right)}_n$	$k!S(n,k)$
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labeled	unlabeled	$\begin{array}{c}S(n,1)+S(n,2)\\ +\mathrm{\cdots }+S(n,k)\end{array}$		$S(n,k)$
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28: 24.20 Tables

§24.20 Tables

… ►Wagstaff (1978) gives complete prime factorizations of $N_n$ and $E_n$ for $n=20(2)60$ and $n=8(2)42$, respectively. …

29: 24.9 Inequalities

§24.9 Inequalities

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24.9.1 $$|B_{2n}|>|B_{2n}\left(x\right)|,$$ $1>x>0$,

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Symbols:

$B_n$: Bernoulli numbers, $B_n\left(x\right)$: Bernoulli polynomials, $n$: integer and $x$: real or complex

A&S Ref:

23.1.13

Referenced by:

§24.9

Permalink:

http://dlmf.nist.gov/24.9.E1

Encodings:

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See also:

Annotations for §24.9 and Ch.24

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24.9.2 $$(2-2^{1-2n})|B_{2n}|\ge |B_{2n}\left(x\right)-B_{2n}|,$$ $1\ge x\ge 0$.

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Symbols:

$B_n$: Bernoulli numbers, $B_n\left(x\right)$: Bernoulli polynomials, $n$: integer and $x$: real or complex

Referenced by:

§24.9

Permalink:

http://dlmf.nist.gov/24.9.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §24.9 and Ch.24

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24.9.3 $$4^{-n}|E_{2n}|>{(-1)}^n{E}_{2n}\left(x\right)>0,$$

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Symbols:

$E_n$: Euler numbers, $E_n\left(x\right)$: Euler polynomials, $n$: integer and $x$: real or complex

A&S Ref:

23.1.13

Referenced by:

§24.9, §24.9

Permalink:

http://dlmf.nist.gov/24.9.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §24.9 and Ch.24

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24.9.6 $$5\sqrt{\pi n}{\left(\frac{n}{\pi \mathrm{e}}\right)}^{2n}>{(-1)}^{n+1}{B}_{2n}>4\sqrt{\pi n}{\left(\frac{n}{\pi \mathrm{e}}\right)}^{2n},$$

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Symbols:

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

Referenced by:

§24.9, §24.9

Permalink:

http://dlmf.nist.gov/24.9.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §24.9 and Ch.24

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30: 27.16 Cryptography

§27.16 Cryptography

… ►The primes are kept secret but their product $n=pq$, an 800-digit number, is made public. …With the most efficient computer techniques devised to date (2010), factoring an 800-digit number may require billions of years on a single computer. For this reason, the codes are considered unbreakable, at least with the current state of knowledge on factoring large numbers. … ►Thus, $y\equiv x^r(modn)$ and . …