哪里能买到安蒂奥克学院文凭毕业证【假证加微aptao168】8e18Buz

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11: 26.3 Lattice Paths: Binomial Coefficients

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Table 26.3.1: Binomial coefficients $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$.

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$m$	$n$
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8	1	8	28	56	70	56	28	8	1
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Table 26.3.2: Binomial coefficients $\left(\genfrac{}{}{0.0pt}{}{m+n}{m}\right)$ for lattice paths.

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$m$	$n$
	0	1	2	3	4	5	6	7	8
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1	1	2	3	4	5	6	7	8	9
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7	1	8	36	120	330	792	1716	3432	6435
8	1	9	45	165	495	1287	3003	6435	12870

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12: 26.13 Permutations: Cycle Notation

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26.13.2

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Referenced by:

§26.13, §26.13, §26.13

Permalink:

http://dlmf.nist.gov/26.13.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §26.13 and Ch.26

►is $\left(1,3,2,5,7\right)\left(4\right)\left(6,8\right)$ in cycle notation. …In consequence, (26.13.2) can also be written as $\left(1,3,2,5,7\right)\left(6,8\right)$. … ►For the example (26.13.2), this decomposition is given by $\left(1,3,2,5,7\right)\left(6,8\right)=\left(1,3\right)\left(2,3\right)\left(2,5\right)\left(5,7\right)\left(6,8\right).$ … ►Again, for the example (26.13.2) a minimal decomposition into adjacent transpositions is given by $\left(1,3,2,5,7\right)\left(6,8\right)=\left(2,3\right)\left(1,2\right)\left(4,5\right)\left(3,4\right)\left(2,3\right)\left(3,4\right)\left(4,5\right)\left(6,7\right)\left(5,6\right)\left(7,8\right)\left(6,7\right)$: $inv(\left(1,3,2,5,7\right)\left(6,8\right))=11$.

13: 7.3 Graphics

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Figure 7.3.4: ${|ℱ\left(x\right)|}^2$, $-8\le x\le 8$. … Magnify

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14: 11.15 Approximations

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Luke (1975, pp. 416–421) gives Chebyshev-series expansions for ${𝐇}_n\left(x\right)$, ${𝐋}_n\left(x\right)$, $0\le \left|x\right|\le 8$, and ${𝐇}_n\left(x\right)-Y_n\left(x\right)$, $x\ge 8$, for $n=0,1$; ${\int }_0^x{t}^{-m}{𝐇}_0\left(t\right)dt$, ${\int }_0^x{t}^{-m}{𝐋}_0\left(t\right)dt$, $0\le \left|x\right|\le 8$, $m=0,1$ and ${\int }_0^x({𝐇}_0\left(t\right)-Y_0\left(t\right))dt$, ${\int }_x^{\mathrm{\infty }}{t}^{-1}({𝐇}_0\left(t\right)-Y_0\left(t\right))dt$, $x\ge 8$; the coefficients are to 20D.

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15: 14 Legendre and Related Functions

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16: 30.17 Tables

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Stratton et al. (1956) tabulates quantities closely related to ${\lambda }_n^m\left({\gamma }^2\right)$ and $a_{n,k}^m({\gamma }^2)$ for $0\le m\le 8$, $m\le n\le 8$, $-64\le {\gamma }^2\le 64$. Precision is 7S.

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EraŠevskaja et al. (1973, 1976) gives $S^{m(j)}(\mathrm{i}y,-\mathrm{i}c)$, $S^{m(j)}(z,\gamma )$ and their first derivatives for $j=1,2$, $0.5\le c\le 8$, $y=0,0.5,1,1.5$, $0.5\le \gamma \le 8$, $z=1.01,1.1,1.4,1.8$. Precision is 15S.

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17: Richard B. Paris

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8 Incomplete Gamma and Related Functions

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18: 25.20 Approximations

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Piessens and Branders (1972) gives the coefficients of the Chebyshev-series expansions of $s\zeta \left(s+1\right)$ and $\zeta \left(s+k\right)$, $k=2,3,4,5,8$, for $0\le s\le 1$ (23D).

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Antia (1993) gives minimax rational approximations for $\mathrm{\Gamma }\left(s+1\right)F_s(x)$, where $F_s(x)$ is the Fermi–Dirac integral (25.12.14), for the intervals and , with $s=-\frac{1}{2},\frac{1}{2},\frac{3}{2},\frac{5}{2}$. For each $s$ there are three sets of approximations, with relative maximum errors ${10}^{-4},{10}^{-8},{10}^{-12}$.

19: 27.13 Functions

… ►Lagrange (1770) proves that $g\left(2\right)=4$, and during the next 139 years the existence of $g\left(k\right)$ was shown for $k=3,4,5,6,7,8,10$. …For example, $g\left(3\right)=9$, $g\left(4\right)=19$, $g\left(5\right)=37$, $g\left(6\right)=73$, $g\left(7\right)=143$, and $g\left(8\right)=279$. … ►Hence $r_2\left(5\right)=8$ because both divisors, $1$ and $5$, are congruent to $1(mod4)$. … ►Explicit formulas for $r_k\left(n\right)$ have been obtained by similar methods for $k=6,8,10$, and $12$, but they are more complicated. …For more than 8 squares, Milne’s identities are not the same as those obtained earlier by Mordell and others.

20: 33.20 Expansions for Small $|ϵ|$

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$f(0,\mathrm{\ell };r)={(2r)}^{1/2}{J}_{2\mathrm{\ell }+1}\left(\sqrt{8r}\right),$

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$h(0,\mathrm{\ell };r)=-{(2r)}^{1/2}{Y}_{2\mathrm{\ell }+1}\left(\sqrt{8r}\right)$ , $r>0$,

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$f(0,\mathrm{\ell };r)={(-1)}^{\mathrm{\ell }+1}{(2|r|)}^{1/2}{I}_{2\mathrm{\ell }+1}\left(\sqrt{8|r|}\right),$

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$h(0,\mathrm{\ell };r)={(-1)}^{\mathrm{\ell }}(2/\pi ){(2|r|)}^{1/2}{K}_{2\mathrm{\ell }+1}\left(\sqrt{8|r|}\right)$ , .

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33.20.4 $${𝖥}_k(\mathrm{\ell };r)=\sum _{p=2k}^{3k}{(2r)}^{(p+1)/2}{C}_{k,p}{J}_{2\mathrm{\ell }+1+p}\left(\sqrt{8r}\right),$$ $r>0$,

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Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $k$: nonnegative integer, $\mathrm{\ell }$: nonnegative integer, $r$: real variable, ${𝖥}_k(\mathrm{\ell };r)$: coefficient and $C_{k,p}$: coefficients

Permalink:

http://dlmf.nist.gov/33.20.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §33.20(ii), §33.20 and Ch.33

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