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1: 26.17 The Twelvefold Way

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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	$k^n$	${\left(k-n+1\right)}_n$	$k!S(n,k)$
unlabeled	labeled	$\left({\displaystyle \genfrac{}{}{0pt}{}{k+n-1}{n}}\right)$	$\left({\displaystyle \genfrac{}{}{0pt}{}{k}{n}}\right)$	$\left({\displaystyle \genfrac{}{}{0pt}{}{n-1}{n-k}}\right)$
labeled	unlabeled	$\begin{array}{c}S(n,1)+S(n,2)\\ +\mathrm{\cdots }+S(n,k)\end{array}$		$S(n,k)$
unlabeled	unlabeled	$p_k\left(n\right)$		$p_k\left(n\right)-p_{k-1}\left(n\right)$

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2: 31.14 General Fuchsian Equation

… ►The general second-order Fuchsian equation with $N+1$ regular singularities at $z=a_j$, $j=1,2,\mathrm{\dots },N$, and at $\mathrm{\infty }$, is given by ►

31.14.1 $$\frac{d^{2}w}{{dz}^2}+\left(\sum _{j=1}^N\frac{{\gamma }_j}{z-a_j}\right)\frac{dw}{dz}+\left(\sum _{j=1}^N\frac{q_j}{z-a_j}\right)w=0,$$ ${\sum }_{j=1}^N{q}_j=0$.

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Symbols:

$\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $z$: complex variable, $\gamma$: real or complex parameter, $j$: nonnegative integer, $a$: complex parameter, $q$: real or complex parameter and $N+1$: number of singularities

Referenced by:

§31.15(i)

Permalink:

http://dlmf.nist.gov/31.14.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §31.14(i), §31.14 and Ch.31

►The exponents at the finite singularities $a_j$ are $\{0,1-{\gamma }_j\}$ and those at $\mathrm{\infty }$ are $\{\alpha ,\beta \}$, where …With $a_1=0$ and $a_2=1$ the total number of free parameters is $3N-3$. … ►

31.14.3 $$w(z)=\left(\prod _{j=1}^N{(z-a_j)}^{-{\gamma }_j/2}\right)W(z),$$

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Symbols:

$z$: complex variable, $\gamma$: real or complex parameter, $j$: nonnegative integer, $a$: complex parameter, $N+1$: number of singularities and $W(z)$: solution

Permalink:

http://dlmf.nist.gov/31.14.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §31.14(i), §31.14(i), §31.14 and Ch.31

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3: 26.5 Lattice Paths: Catalan Numbers

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26.5.1 $$C\left(n\right)=\frac{1}{n+1}\left(\genfrac{}{}{0.0pt}{}{2n}{n}\right)=\frac{1}{2n+1}\left(\genfrac{}{}{0.0pt}{}{2n+1}{n}\right)=\left(\genfrac{}{}{0.0pt}{}{2n}{n}\right)-\left(\genfrac{}{}{0.0pt}{}{2n}{n-1}\right)=\left(\genfrac{}{}{0.0pt}{}{2n-1}{n}\right)-\left(\genfrac{}{}{0.0pt}{}{2n-1}{n+1}\right).$$

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Defines:

$C\left(n\right)$: Catalan number

Symbols:

$\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient and $n$: nonnegative integer

Referenced by:

§26.5(iv)

Permalink:

http://dlmf.nist.gov/26.5.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §26.5(i), §26.5 and Ch.26

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26.5.3 $$C\left(n+1\right)=\sum _{k=0}^{n}C\left(k\right)C\left(n-k\right),$$

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Symbols:

$C\left(n\right)$: Catalan number, $k$: nonnegative integer and $n$: nonnegative integer

Permalink:

http://dlmf.nist.gov/26.5.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §26.5(iii), §26.5 and Ch.26

►

26.5.4 $$C\left(n+1\right)=\frac{2(2n+1)}{n+2}C\left(n\right),$$

ⓘ

Symbols:

$C\left(n\right)$: Catalan number and $n$: nonnegative integer

Permalink:

http://dlmf.nist.gov/26.5.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §26.5(iii), §26.5 and Ch.26

►

26.5.5 $$C\left(n+1\right)=\sum _{k=0}^{\lfloor n/2\rfloor }\left(\genfrac{}{}{0.0pt}{}{n}{2k}\right)2^{n-2k}C\left(k\right).$$

ⓘ

Symbols:

$C\left(n\right)$: Catalan number, $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, $\lfloor x\rfloor$: floor of $x$ , $k$: nonnegative integer and $n$: nonnegative integer

Permalink:

http://dlmf.nist.gov/26.5.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §26.5(iii), §26.5 and Ch.26

… ►

26.5.7 $$\underset{n\to \mathrm{\infty }}{lim}\frac{C\left(n+1\right)}{C\left(n\right)}=4.$$

ⓘ

Symbols:

$C\left(n\right)$: Catalan number and $n$: nonnegative integer

Permalink:

http://dlmf.nist.gov/26.5.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §26.5(iv), §26.5 and Ch.26

4: 31.15 Stieltjes Polynomials

… ►If $t_k$ is a zero of the Van Vleck polynomial $V(z)$, corresponding to an $n$th degree Stieltjes polynomial $S(z)$, and $z_1^{\prime },z_2^{\prime },\mathrm{\dots },z_{n-1}^{\prime }$ are the zeros of $S^{\prime }(z)$ (the derivative of $S(z)$), then $t_k$ is either a zero of $S^{\prime }(z)$ or a solution of the equation … ►The zeros $z_k$, $k=1,2,\mathrm{\dots },n$, of the Stieltjes polynomial $S(z)$ are the critical points of the function $G$, that is, points at which $\partial G/\partial {\zeta }_k=0$, $k=1,2,\mathrm{\dots },n$, where … ►then there are exactly $\left(\genfrac{}{}{0.0pt}{}{n+N-2}{N-2}\right)$ polynomials $S(z)$, each of which corresponds to each of the $\left(\genfrac{}{}{0.0pt}{}{n+N-2}{N-2}\right)$ ways of distributing its $n$ zeros among $N-1$ intervals $(a_j,a_{j+1})$, $j=1,2,\mathrm{\dots },N-1$. … ►If the exponent and singularity parameters satisfy (31.15.5)–(31.15.6), then for every multi-index $𝐦=(m_1,m_2,\mathrm{\dots },m_{N-1})$, where each $m_j$ is a nonnegative integer, there is a unique Stieltjes polynomial with $m_j$ zeros in the open interval $(a_j,a_{j+1})$ for each $j=1,2,\mathrm{\dots },N-1$. … ►Let $S_{𝐦}(z)$ and $S_{𝐥}(z)$ be Stieltjes polynomials corresponding to two distinct multi-indices $𝐦=(m_1,m_2,\mathrm{\dots },m_{N-1})$ and $𝐥=({\mathrm{\ell }}_1,{\mathrm{\ell }}_2,\mathrm{\dots },{\mathrm{\ell }}_{N-1})$. …

5: 26.8 Set Partitions: Stirling Numbers

… ► $s(n,k)$ denotes the Stirling number of the first kind: ${(-1)}^{n-k}$ times the number of permutations of $\{1,2,\mathrm{\dots },n\}$ with exactly $k$ cycles. … ►where ${\left(x\right)}_n$ is the Pochhammer symbol: $x(x+1)\mathrm{\cdots }(x+n-1)$. … ►For $n\ge 1$, … ►uniformly for $n=o\left(k^{1/2}\right)$. ►For asymptotic approximations for $s(n+1,k+1)$ and $S(n,k)$ that apply uniformly for $1\le k\le n$ as $n\to \mathrm{\infty }$ see Temme (1993) and Temme (2015, Chapter 34). …

6: 24.5 Recurrence Relations

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24.5.1 $$\sum _{k=0}^{n-1}\left(\genfrac{}{}{0pt}{}{n}{k}\right)B_k\left(x\right)=n{x}^{n-1},$$ $n=2,3,\mathrm{\dots }$,

ⓘ

Symbols:

$B_n\left(x\right)$: Bernoulli polynomials, $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, $k$: integer, $n$: integer and $x$: real or complex

Permalink:

http://dlmf.nist.gov/24.5.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §24.5(i), §24.5 and Ch.24

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24.5.5 $$\sum _{k=0}^n\left(\genfrac{}{}{0pt}{}{n}{k}\right)2^k{E}_{n-k}+E_n=2.$$

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Symbols:

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

Permalink:

http://dlmf.nist.gov/24.5.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §24.5(i), §24.5 and Ch.24

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24.5.6 $$\sum _{k=2}^n\left(\genfrac{}{}{0pt}{}{n}{k-2}\right)\frac{B_k}{k}=\frac{1}{(n+1)(n+2)}-B_{n+1},$$ $n=2,3,\mathrm{\dots }$,

ⓘ

Symbols:

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

Referenced by:

§24.5(ii)

Permalink:

http://dlmf.nist.gov/24.5.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §24.5(ii), §24.5 and Ch.24

►

24.5.7 $$\sum _{k=0}^n\left(\genfrac{}{}{0pt}{}{n}{k}\right)\frac{B_k}{n+2-k}=\frac{B_{n+1}}{n+1},$$ $n=1,2,\mathrm{\dots }$,

ⓘ

Symbols:

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

Referenced by:

§24.5(ii)

Permalink:

http://dlmf.nist.gov/24.5.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §24.5(ii), §24.5 and Ch.24

►

24.5.8 $$\sum _{k=0}^n\frac{2^{2k}{B}_{2k}}{(2k)!(2n+1-2k)!}=\frac{1}{(2n)!},$$ $n=1,2,\mathrm{\dots }$.

ⓘ

Symbols:

$B_n$: Bernoulli numbers, $!$: factorial (as in $n!$), $k$: integer and $n$: integer

Referenced by:

§24.5(ii)

Permalink:

http://dlmf.nist.gov/24.5.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §24.5(ii), §24.5 and Ch.24

…

7: Customize DLMF

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8: 26.6 Other Lattice Path Numbers

… ► $D(m,n)$ is the number of paths from $(0,0)$ to $(m,n)$ that are composed of directed line segments of the form $(1,0)$, $(0,1)$, or $(1,1)$. … ► $M(n)$ is the number of lattice paths from $(0,0)$ to $(n,n)$ that stay on or above the line $y=x$ and are composed of directed line segments of the form $(2,0)$, $(0,2)$, or $(1,1)$. … ► $N(n,k)$ is the number of lattice paths from $(0,0)$ to $(n,n)$ that stay on or above the line $y=x$, are composed of directed line segments of the form $(1,0)$ or $(0,1)$, and for which there are exactly $k$ occurrences at which a segment of the form $(0,1)$ is followed by a segment of the form $(1,0)$. … ► $r(n)$ is the number of paths from $(0,0)$ to $(n,n)$ that stay on or above the diagonal $y=x$ and are composed of directed line segments of the form $(1,0)$, $(0,1)$, or $(1,1)$. … ►

26.6.10 $$D(m,n)=D(m,n-1)+D(m-1,n)+D(m-1,n-1),$$ $m,n\ge 1$,

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Symbols:

$m$: nonnegative integer, $n$: nonnegative integer and $D(m,n)$: Delannoy number

Referenced by:

§26.6(iii)

Permalink:

http://dlmf.nist.gov/26.6.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §26.6(iii), §26.6 and Ch.26

…

9: 24.15 Related Sequences of Numbers

… ►

24.15.7 $$B_n=\sum _{k=0}^n{(-1)}^k\left(\genfrac{}{}{0.0pt}{}{n+1}{k+1}\right)S(n+k,k)/\left(\genfrac{}{}{0.0pt}{}{n+k}{k}\right),$$

ⓘ

Symbols:

$B_n$: Bernoulli numbers, $S(n,k)$: Stirling number of the second kind, $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, $k$: integer and $n$: integer

Referenced by:

§24.15(iii)

Permalink:

http://dlmf.nist.gov/24.15.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §24.15(iii), §24.15 and Ch.24

►

24.15.8 $$\sum _{k=0}^n{(-1)}^{n+k}s(n+1,k+1)B_k=\frac{n!}{n+1}.$$

ⓘ

Symbols:

$B_n$: Bernoulli numbers, $s(n,k)$: Stirling number of the first kind, $!$: factorial (as in $n!$), $k$: integer and $n$: integer

Referenced by:

§24.15(iii)

Permalink:

http://dlmf.nist.gov/24.15.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §24.15(iii), §24.15 and Ch.24

… ►

24.15.10 $$\frac{2n-1}{4n}{p}^2{B}_{2n}\equiv S(p+2n,p-1)(mod{p}^3),$$ $2\le 2n\le p-3$.

ⓘ

Symbols:

$B_n$: Bernoulli numbers, $S(n,k)$: Stirling number of the second kind, $\equiv$: modular equivalence, $n$: integer and $p$: prime

Referenced by:

§24.15(iii)

Permalink:

http://dlmf.nist.gov/24.15.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §24.15(iii), §24.15 and Ch.24

… ►The Fibonacci numbers are defined by $u_0=0$, $u_1=1$, and $u_{n+1}=u_n+u_{n-1}$, $n\ge 1$. The Lucas numbers are defined by $v_0=2$, $v_1=1$, and $v_{n+1}=v_n+v_{n-1}$, $n\ge 1$. …

10: 26.11 Integer Partitions: Compositions

… ►For example, there are eight compositions of 4: $4,3+1,1+3,2+2,2+1+1,1+2+1,1+1+2$, and $1+1+1+1$. …$c(\in T,n)$ is the number of compositions of $n$ with no 1’s, where again $T=\{2,3,4,\mathrm{\dots }\}$. … ►

$F_1=1$ ,

►

$F_n=F_{n-1}+F_{n-2}$ , $n\ge 2$.

… ►

26.11.7 $$F_n=\frac{{(1+\sqrt{5})}^n-{(1-\sqrt{5})}^n}{2^n\sqrt{5}}.$$

ⓘ

Symbols:

$n$: nonnegative integer and $F_n$: Fibonacci numbers

Referenced by:

(18.5.4_5)

Permalink:

http://dlmf.nist.gov/26.11.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §26.11 and Ch.26

…