# Narayana numbers

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## 21—30 of 213 matching pages

##### 21: How to Cite
When referring to a specific item (chapter, section, equation, …), please mention the reference number. …Your readers can then readily locate the item within the book or online (at http://dlmf.nist.gov/5.4.E1 in this example; see Permalinks & Reference numbers). … Since the reference number is immediately recognized from the URL as Eq. …
###### Permalinks & Reference numbers
The following table outlines the correspondence between reference numbers as they appear in the Handbook, and the URL’s that find the same item online. …
##### 22: 24.6 Explicit Formulas
###### §24.6 Explicit Formulas
24.6.2 $B_{n}=\frac{1}{n+1}\sum_{k=1}^{n}\sum_{j=1}^{k}(-1)^{j}j^{n}{\genfrac{(}{)}{0.% 0pt}{}{n+1}{k-j}}\Bigg{/}{\genfrac{(}{)}{0.0pt}{}{n}{k}},$
##### 23: 27.13 Functions
###### §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. …
##### 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 $(2|p)=(-1)^{(p^{2}-1)/8}.$
If $p,q$ are distinct odd primes, then the quadratic reciprocity law states that … If an odd integer $P$ has prime factorization $P=\prod_{r=1}^{\nu\left(n\right)}p^{a_{r}}_{r}$, then the Jacobi symbol $(n|P)$ is defined by $(n|P)=\prod_{r=1}^{\nu\left(n\right)}{(n|p_{r})}^{a_{r}}$, with $(n|1)=1$. …
##### 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 $S\left(n,k\right)$ and $p_{k}\left(n\right)$ are defined in §§26.8(i) and 26.9(i). …
##### 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
24.9.1 $|B_{2n}|>|B_{2n}\left(x\right)|,$ $1>x>0$,
24.9.2 $(2-2^{1-2n})|B_{2n}|\geq|B_{2n}\left(x\right)-B_{2n}|,$ $1\geq x\geq 0$.
24.9.6 $5\sqrt{\pi n}\left(\frac{n}{\pi e}\right)^{2n}>(-1)^{n+1}B_{2n}>4\sqrt{\pi n}% \left(\frac{n}{\pi e}\right)^{2n},$
##### 30: 4.19 Maclaurin Series and Laurent Series
In (4.19.3)–(4.19.9), $B_{n}$ are the Bernoulli numbers and $E_{n}$ are the Euler numbers (§§24.2(i)24.2(ii)).
4.19.3 $\tan z=z+\frac{z^{3}}{3}+\frac{2}{15}z^{5}+\frac{17}{315}z^{7}+\cdots+\frac{(-% 1)^{n-1}2^{2n}(2^{2n}-1)B_{2n}}{(2n)!}z^{2n-1}+\cdots,$ $|z|<\frac{1}{2}\pi$,
4.19.4 $\csc z=\frac{1}{z}+\frac{z}{6}+\frac{7}{360}z^{3}+\frac{31}{15120}z^{5}+\cdots% +\frac{(-1)^{n-1}2(2^{2n-1}-1)B_{2n}}{(2n)!}z^{2n-1}+\cdots,$ $0<|z|<\pi$,
4.19.6 $\cot z=\frac{1}{z}-\frac{z}{3}-\frac{z^{3}}{45}-\frac{2}{945}z^{5}-\cdots-% \frac{(-1)^{n-1}2^{2n}B_{2n}}{(2n)!}z^{2n-1}-\cdots,$ $0<|z|<\pi$,