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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31—40 of 253 matching pages

$n$	$p_{n}$	$p_{n+10}$	$p_{n+20}$	$p_{n+30}$	$p_{n+40}$	$p_{n+50}$	$p_{n+60}$	$p_{n+70}$	$p_{n+80}$	$p_{n+90}$
…
6	13	53	101	151	199	263	317	383	443	503
…

$n$	$\phi\left(n\right)$	$d\left(n\right)$	$\sigma\left(n\right)$	$n$	$\phi\left(n\right)$	$d\left(n\right)$	$\sigma\left(n\right)$	$n$	$\phi\left(n\right)$	$d\left(n\right)$	$\sigma\left(n\right)$	$n$	$\phi\left(n\right)$	$d\left(n\right)$	$\sigma\left(n\right)$
1	1	1	1	14	6	4	24	27	18	4	40	40	16	8	90
…
5	4	2	6	18	6	6	39	31	30	2	32	44	20	6	84
6	2	4	12	19	18	2	20	32	16	6	63	45	24	6	78
7	6	2	8	20	8	6	42	33	20	4	48	46	22	4	72
…

32: 30.10 Series and Integrals

… ►For expansions in products of spherical Bessel functions, see Flammer (1957, Chapter 6). …

33: 34.6 Definition: $\mathit{9j}$ Symbol

… ►The

\mathit{9j}

symbol may be defined either in terms of

\mathit{3j}

symbols or equivalently in terms of

\mathit{6j}

symbols: ►

34.6.1

\begin{Bmatrix}j_{11}&j_{12}&j_{13}\\ j_{21}&j_{22}&j_{23}\\ j_{31}&j_{32}&j_{33}\end{Bmatrix}=\sum_{\mbox{\scriptsize all }m_{rs}}\begin{% pmatrix}j_{11}&j_{12}&j_{13}\\ m_{11}&m_{12}&m_{13}\end{pmatrix}\begin{pmatrix}j_{21}&j_{22}&j_{23}\\ m_{21}&m_{22}&m_{23}\end{pmatrix}\begin{pmatrix}j_{31}&j_{32}&j_{33}\\ m_{31}&m_{32}&m_{33}\end{pmatrix}\*\begin{pmatrix}j_{11}&j_{21}&j_{31}\\ m_{11}&m_{21}&m_{31}\end{pmatrix}\begin{pmatrix}j_{12}&j_{22}&j_{32}\\ m_{12}&m_{22}&m_{32}\end{pmatrix}\begin{pmatrix}j_{13}&j_{23}&j_{33}\\ m_{13}&m_{23}&m_{33}\end{pmatrix},

ⓘ

Defines:: $\begin{Bmatrix}\NVar{j_{11}}&\NVar{j_{12}}&\NVar{j_{13}}\\ \NVar{j_{21}}&\NVar{j_{22}}&\NVar{j_{23}}\\ \NVar{j_{31}}&\NVar{j_{32}}&\NVar{j_{33}}\end{Bmatrix}$ : $\mathit{9j}$ symbol
Symbols:: $\begin{pmatrix}\NVar{j_{1}}&\NVar{j_{2}}&\NVar{j_{3}}\\ \NVar{m_{1}}&\NVar{m_{2}}&\NVar{m_{3}}\end{pmatrix}$ : $\mathit{3j}$ symbol, $j,j_{r}$ : non-negative integers or non-negative integers plus one half., $r$ : nonnegative integer and $s$ : integer
Permalink:: http://dlmf.nist.gov/34.6.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §34.6 and Ch.34

►

34.6.2

\begin{Bmatrix}j_{11}&j_{12}&j_{13}\\ j_{21}&j_{22}&j_{23}\\ j_{31}&j_{32}&j_{33}\end{Bmatrix}=\sum_{j}(-1)^{2j}(2j+1)\begin{Bmatrix}j_{11}% &j_{21}&j_{31}\\ j_{32}&j_{33}&j\end{Bmatrix}\begin{Bmatrix}j_{12}&j_{22}&j_{32}\\ j_{21}&j&j_{23}\end{Bmatrix}\begin{Bmatrix}j_{13}&j_{23}&j_{33}\\ j&j_{11}&j_{12}\end{Bmatrix}.

ⓘ

Symbols:: $\begin{Bmatrix}\NVar{j_{11}}&\NVar{j_{12}}&\NVar{j_{13}}\\ \NVar{j_{21}}&\NVar{j_{22}}&\NVar{j_{23}}\\ \NVar{j_{31}}&\NVar{j_{32}}&\NVar{j_{33}}\end{Bmatrix}$ : $\mathit{9j}$ symbol, $\begin{Bmatrix}\NVar{j_{1}}&\NVar{j_{2}}&\NVar{j_{3}}\\ \NVar{l_{1}}&\NVar{l_{2}}&\NVar{l_{3}}\end{Bmatrix}$ : $\mathit{6j}$ symbol and $j,j_{r}$ : non-negative integers or non-negative integers plus one half.
Referenced by:: §34.13
Permalink:: http://dlmf.nist.gov/34.6.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §34.6 and Ch.34

…

… ►He has been an Associate Professor at Moscow State University (Russia), an Ostrowski Fellow at Institut Henri Poincaré and Université Paris 6 (France), a Humboldt Fellow at the Cologne University (Germany) and a Visiting Researcher at the Max Planck Institute for Mathematics in Bonn (Germany). …

36: 26.9 Integer Partitions: Restricted Number and Part Size

… ►

Table 26.9.1: Partitions

p_{k}\left(n\right)

► ►►►►►

$n$	$k$
$n$	0	1	2	3	4	5	6	7	8	9	10
…
5	0	1	3	5	6	7	7	7	7	7	7
6	0	1	4	7	9	10	11	11	11	11	11
…

► … ►The conjugate to the example in Figure 26.9.1 is

6+5+4+2+1+1+1

. … ►

p_{3}\left(n\right)=1+\left\lfloor\frac{n^{2}+6n}{12}\right\rfloor.

…

37: 30.9 Asymptotic Approximations and Expansions

… ►

2^{6}\beta_{1}=-q^{3}-11q+32m^{2}q,

… ►

2^{16}\beta_{4}=-63q^{6}-4940q^{4}-43327q^{2}-22470+128m^{2}(115q^{4}+1310q^{2% }+735)-24576m^{4}(q^{2}+1),

►

2^{20}\beta_{5}=-527q^{7}-61529q^{5}-10\;43961q^{3}-22\;41599q+32m^{2}(5739q^{% 5}+1\;27550q^{3}+2\;98951q)-2048m^{4}(355q^{3}+1505q)+65536m^{6}q.

… ►

2^{6}c_{2}=-5q^{4}-10q^{2}-1+2m^{2}(3q^{2}+1)-m^{4},

… ►

2^{10}c_{4}=-63q^{6}-340q^{4}-239q^{2}-14+10m^{2}(10q^{4}+23q^{2}+3)-3m^{4}(13% q^{2}+6)+2m^{6},

…

38: 26.21 Tables

… ►It also contains a table of Gaussian polynomials up to

\genfrac{[}{]}{0.0pt}{}{12}{6}_{q}

. …

39: 29.7 Asymptotic Expansions

… ►

29.7.4

\tau_{1}=\frac{p}{2^{6}}((1+k^{2})^{2}(p^{2}+3)-4k^{2}(p^{2}+5)).

ⓘ

Symbols:: $p$ : nonnegative integer, $k$ : real parameter and $\tau_{j}$ : coefficients
Permalink:: http://dlmf.nist.gov/29.7.E4
Encodings:: pMML, png, TeX
See also:: Annotations for §29.7(i), §29.7 and Ch.29

… ►

29.7.8

\tau_{4}=\frac{1}{2^{16}}((1+k^{2})^{5}(63p^{6}+1260p^{4}+2943p^{2}+486)-8k^{2% }(1+k^{2})^{3}(49p^{6}+1010p^{4}+2493p^{2}+432)+16k^{4}(1+k^{2})(35p^{6}+760p^% {4}+2043p^{2}+378)).

ⓘ

Symbols:: $p$ : nonnegative integer, $k$ : real parameter and $\tau_{j}$ : coefficients
Permalink:: http://dlmf.nist.gov/29.7.E8
Encodings:: pMML, png, TeX
See also:: Annotations for §29.7(i), §29.7 and Ch.29

…

40: 32.13 Reductions of Partial Differential Equations

… ►

32.13.1

v_{t}-6v^{2}v_{x}+v_{xxx}=0,

ⓘ

Symbols:: $v(x,t)$ : solution
Permalink:: http://dlmf.nist.gov/32.13.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §32.13(i), §32.13 and Ch.32

… ►

32.13.3

u_{t}+6uu_{x}+u_{xxx}=0,

ⓘ

Symbols:: $u(x,t)$ : solution
Referenced by:: §32.13(i)
Permalink:: http://dlmf.nist.gov/32.13.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §32.13(i), §32.13 and Ch.32

… ►

32.13.8

u_{tt}=u_{xx}-6(u^{2})_{xx}+u_{xxxx},

ⓘ

Symbols:: $u(x,t)$ : solution
Permalink:: http://dlmf.nist.gov/32.13.E8
Encodings:: pMML, png, TeX
See also:: Annotations for §32.13(iii), §32.13 and Ch.32

… ►

32.13.10

v^{\prime\prime}=6v^{2}+(c^{2}-1)v+Az+B,

ⓘ

Symbols:: $z$ : real, $c$ : constant, $v(z)$ : solution, $A$ : integration constant and $B$ : integration constant
Permalink:: http://dlmf.nist.gov/32.13.E10
Encodings:: pMML, png, TeX
See also:: Annotations for §32.13(iii), §32.13 and Ch.32

…

长沙麻将玩法二五八起手胡【信誉下注平台qee9.com】6pFGaMrr

31—40 of 253 matching pages

31: 27.2 Functions

32: 30.10 Series and Integrals

33: 34.6 Definition: $\mathit{9j}$ Symbol

34: Leonard C. Maximon

35: Wadim Zudilin

36: 26.9 Integer Partitions: Restricted Number and Part Size

37: 30.9 Asymptotic Approximations and Expansions

38: 26.21 Tables

39: 29.7 Asymptotic Expansions

40: 32.13 Reductions of Partial Differential Equations

$n$	$\phi\left(n\right)$	$d\left(n\right)$	$\sigma\left(n\right)$	$n$	$\phi\left(n\right)$	$d\left(n\right)$	$\sigma\left(n\right)$	$n$	$\phi\left(n\right)$	$d\left(n\right)$	$\sigma\left(n\right)$	$n$	$\phi\left(n\right)$	$d\left(n\right)$	$\sigma\left(n\right)$
1	1	1	1	14	6	4	24	27	18	4	40	40	16	8	90
…
5	4	2	6	18	6	6	39	31	30	2	32	44	20	6	84
6	2	4	12	19	18	2	20	32	16	6	63	45	24	6	78
7	6	2	8	20	8	6	42	33	20	4	48	46	22	4	72
…

$n$	$\phi\left(n\right)$	$d\left(n\right)$	$\sigma\left(n\right)$	$n$	$\phi\left(n\right)$	$d\left(n\right)$	$\sigma\left(n\right)$	$n$	$\phi\left(n\right)$	$d\left(n\right)$	$\sigma\left(n\right)$	$n$	$\phi\left(n\right)$	$d\left(n\right)$	$\sigma\left(n\right)$
1	1	1	1	14	6	4	24	27	18	4	40	40	16	8	90
…
5	4	2	6	18	6	6	39	31	30	2	32	44	20	6	84
6	2	4	12	19	18	2	20	32	16	6	63	45	24	6	78
7	6	2	8	20	8	6	42	33	20	4	48	46	22	4	72
…

长沙麻将玩法二五八起手胡【信誉下注平台qee9.com】6pFGaMrr

31—40 of 253 matching pages

31: 27.2 Functions

32: 30.10 Series and Integrals

33: 34.6 Definition: 9 ⁢ j Symbol

34: Leonard C. Maximon

35: Wadim Zudilin

36: 26.9 Integer Partitions: Restricted Number and Part Size

37: 30.9 Asymptotic Approximations and Expansions

38: 26.21 Tables

39: 29.7 Asymptotic Expansions

40: 32.13 Reductions of Partial Differential Equations

33: 34.6 Definition: $\mathit{9j}$ Symbol

$n$	$\phi\left(n\right)$	$d\left(n\right)$	$\sigma\left(n\right)$	$n$	$\phi\left(n\right)$	$d\left(n\right)$	$\sigma\left(n\right)$	$n$	$\phi\left(n\right)$	$d\left(n\right)$	$\sigma\left(n\right)$	$n$	$\phi\left(n\right)$	$d\left(n\right)$	$\sigma\left(n\right)$
1	1	1	1	14	6	4	24	27	18	4	40	40	16	8	90
…
5	4	2	6	18	6	6	39	31	30	2	32	44	20	6	84
6	2	4	12	19	18	2	20	32	16	6	63	45	24	6	78
7	6	2	8	20	8	6	42	33	20	4	48	46	22	4	72
…