长沙麻将玩法二五八起手胡【信誉下注平台qee9.com】6pFGaMrr

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21: 26.2 Basic Definitions

… ►If, for example, a permutation of the integers 1 through 6 is denoted by

256413

, then the cycles are

{\left(1,2,5\right)}

{\left(3,6\right)}

, and

{\left(4\right)}

. …The function

\sigma

also interchanges 3 and 6, and sends 4 to itself. … ►As an example,

\{1,3,4\}

\{2,6\}

\{5\}

is a partition of

\{1,2,3,4,5,6\}

. … ►

Table 26.2.1: Partitions

p\left(n\right)

► ►►►

$n$	$p\left(n\right)$	$n$	$p\left(n\right)$	$n$	$p\left(n\right)$
…
6	11	23	1255	40	37338
…

►

22: 24.2 Definitions and Generating Functions

… ►

Table 24.2.3: Bernoulli numbers

B_{n}=N/D

► ►►►►►►

$n$	$N$	$D$	$B_{n}$
…
$2$	$1$	$6$	$1.66666\;6667$	$\times 10^{-1}$
…
$6$	$1$	$42$	$2.38095\;2381$	$\times 10^{-2}$
…
$14$	$7$	$6$	$1.16666\;6667$
…
$26$	$85\;53103$	$6$	$1.42551\;7167$	$\times 10^{6}$
…

► ►

Table 24.2.4: Euler numbers

E_{n}

► ►►►

$n$	$E_{n}$
…
$6$	$-61$
…

► …

23: 26.3 Lattice Paths: Binomial Coefficients

… ►

Table 26.3.1: Binomial coefficients

\genfrac{(}{)}{0.0pt}{}{m}{n}

► ►►►►

$m$	$n$
$m$	…
4	1	4	6	4	1
…
6	1	6	15	20	15	6	1
…

► ►

Table 26.3.2: Binomial coefficients

\genfrac{(}{)}{0.0pt}{}{m+n}{m}

for lattice paths.

► ►►►►►

$m$	$n$
$m$	0	1	2	3	4	5	6	7	8
…
1	1	2	3	4	5	6	7	8	9
2	1	3	6	10	15	21	28	36	45
…

► …

24: 6.8 Inequalities

… ►

6.8.3

\frac{x(x+3)}{x^{2}+4x+2}<xe^{x}E_{1}\left(x\right)<\frac{x^{2}+5x+2}{x^{2}+6x% +6}.

ⓘ

Symbols:: $\mathrm{e}$ : base of natural logarithm, $E_{1}\left(\NVar{z}\right)$ : exponential integral and $x$ : real variable
Referenced by:: §6.8
Permalink:: http://dlmf.nist.gov/6.8.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §6.8 and Ch.6

25: Nico M. Temme

… ►

6 Exponential, Logarithmic, Sine, and Cosine Integrals

… ►In November 2015, Temme was named Senior Associate Editor of the DLMF and Associate Editor for Chapters 3, 6, 7, and 12.

26: 3.4 Differentiation

… ►

B_{-1}^{3}=-\tfrac{1}{6}(2-6t+3t^{2}),

… ►

B_{-3}^{6}=-\tfrac{1}{720}(12-8t-45t^{2}+20t^{3}+15t^{4}-6t^{5}),

… ►

B_{-1}^{6}=-\tfrac{1}{48}(36-72t-39t^{2}+52t^{3}+5t^{4}-6t^{5}),

… ►

B_{1}^{6}=\tfrac{1}{48}(36+72t-39t^{2}-52t^{3}+5t^{4}+6t^{5}),

… ►

B_{3}^{6}=\tfrac{1}{720}(12+8t-45t^{2}-20t^{3}+15t^{4}+6t^{5}).

…

27: 32.3 Graphics

… ►Plots of solutions

w_{k}(x)

\mbox{P}_{\mbox{\scriptsize I}}

with

w_{k}(0)=0

and

w_{k}^{\prime}(0)=k

for various values of

k

, and the parabola

6w^{2}+x=0

. … ►

See accompanying text — Figure 32.3.1: $w_{k}(x)$ for $-12\leq x\leq 1.33$ and $k=0.5$ , $0.75$ , $1$ , $1.25$ , and the parabola $6w^{2}+x=0$ , shown in black. Magnify

… ►

►

… ►

32.3.6

u^{2}=-\tfrac{1}{3}x\pm\tfrac{1}{6}\sqrt{x^{2}+12\nu+6}.

ⓘ

Symbols:: $x$ : real, $\nu$ : parameter and $u_{k}(x;\nu)$ : solution
Permalink:: http://dlmf.nist.gov/32.3.E6
Encodings:: pMML, png, TeX
See also:: Annotations for §32.3(iii), §32.3 and Ch.32

…

28: 28.6 Expansions for Small $q$

… ►Leading terms of the power series for

a_{m}\left(q\right)

and

b_{m}\left(q\right)

for

m\leq 6

are: ►

28.6.1

a_{0}\left(q\right)=-\tfrac{1}{2}q^{2}+\tfrac{7}{128}q^{4}-\tfrac{29}{2304}q^{% 6}+\tfrac{68687}{188\;74368}q^{8}+\cdots,

ⓘ

Symbols:: $a_{\NVar{n}}\left(\NVar{q}\right)$ : eigenvalues of Mathieu equation and $q=h^{2}$ : parameter
A&S Ref:: 20.2.25
Referenced by:: §28.6(i)
Permalink:: http://dlmf.nist.gov/28.6.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §28.6(i), §28.6 and Ch.28

… ►

28.6.4

a_{2}\left(q\right)=4+\tfrac{5}{12}q^{2}-\tfrac{763}{13824}q^{4}+\tfrac{10\;02% 401}{796\;26240}q^{6}-\tfrac{16690\;68401}{45\;86471\;42400}q^{8}+\cdots,

ⓘ

Symbols:: $a_{\NVar{n}}\left(\NVar{q}\right)$ : eigenvalues of Mathieu equation and $q=h^{2}$ : parameter
Permalink:: http://dlmf.nist.gov/28.6.E4
Encodings:: pMML, png, TeX
See also:: Annotations for §28.6(i), §28.6 and Ch.28

… ►

28.6.12

a_{6}\left(q\right)=36+\tfrac{1}{70}q^{2}+\tfrac{187}{439\;04000}q^{4}+\tfrac{% 67\;43617}{9293\;59872\;00000}q^{6}+\cdots,

ⓘ

Symbols:: $a_{\NVar{n}}\left(\NVar{q}\right)$ : eigenvalues of Mathieu equation and $q=h^{2}$ : parameter
Permalink:: http://dlmf.nist.gov/28.6.E12
Encodings:: pMML, png, TeX
See also:: Annotations for §28.6(i), §28.6 and Ch.28

►

28.6.13

b_{6}\left(q\right)=36+\tfrac{1}{70}q^{2}+\tfrac{187}{439\;04000}q^{4}-\tfrac{% 58\;61633}{9293\;59872\;00000}q^{6}+\cdots.

ⓘ

Symbols:: $b_{\NVar{n}}\left(\NVar{q}\right)$ : eigenvalues of Mathieu equation and $q=h^{2}$ : parameter
Permalink:: http://dlmf.nist.gov/28.6.E13
Encodings:: pMML, png, TeX
See also:: Annotations for §28.6(i), §28.6 and Ch.28

…

29: 10 Bessel Functions

…

30: 18 Orthogonal Polynomials

…