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11: 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 $1+1+1+1$. $c\left(n\right)$ denotes the number of compositions of $n$, and $c_m\left(n\right)$ is the number of compositions into exactly $m$ parts. $c(\in T,n)$ is the number of compositions of $n$ with no 1’s, where again $T=\{2,3,4,\mathrm{\dots }\}$. … ►The Fibonacci numbers are determined recursively by … ►Additional information on Fibonacci numbers can be found in Rosen et al. (2000, pp. 140–145).

12: 26.5 Lattice Paths: Catalan Numbers

… ►

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

ⓘ

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

… ►

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

ⓘ

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

13: Ronald F. Boisvert

… ►His research interests include numerical solution of partial differential equations, mathematical software, and information services that support computational science. … ►Department of Commerce Gold Medal for Distinguished Achievement in the Federal Service in 2011, and an Outstanding Alumni Award from the Purdue University Department of Computer Science in 2012. …

14: 27.18 Methods of Computation: Primes

… ►These algorithms are used for testing primality of Mersenne numbers, $2^n-1$, and Fermat numbers, $2^{2^n}+1$. …

15: 24.15 Related Sequences of Numbers

… ►

24.15.6 $$B_n=\sum _{k=0}^n{(-1)}^k\frac{k!S(n,k)}{k+1},$$

ⓘ

Symbols:

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

Referenced by:

§24.15(iii)

Permalink:

http://dlmf.nist.gov/24.15.E6

Encodings:

pMML, png, TeX

See also:

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

►

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

… ►

24.15.9 $$p\frac{B_n}{n}\equiv S(p-1+n,p-1)(mod{p}^2),$$ $1\le n\le p-2$,

ⓘ

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.E9

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 $u_{n+1}=u_n+u_{n-1}$, $n\ge 1$. …

16: 26.6 Other Lattice Path Numbers

… ►

26.6.2 $$M(n)=\sum _{k=0}^n\frac{{(-1)}^k}{n+2-k}\left(\genfrac{}{}{0.0pt}{}{n}{k}\right)\left(\genfrac{}{}{0.0pt}{}{2n+2-2k}{n+1-k}\right).$$

ⓘ

Defines:

$M(n)$: Motzkin number (locally)

Symbols:

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

Permalink:

http://dlmf.nist.gov/26.6.E2

Encodings:

pMML, png, TeX

See also:

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

… ►

26.6.4 $$r(n)=D(n,n)-D(n+1,n-1),$$ $n\ge 1$.

ⓘ

Defines:

$r(n)$: Schröder number (locally)

Symbols:

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

Referenced by:

§26.6(i)

Permalink:

http://dlmf.nist.gov/26.6.E4

Encodings:

pMML, png, TeX

See also:

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

… ►

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

ⓘ

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

►

26.6.11 $$M(n)=M(n-1)+\sum _{k=2}^{n}M(k-2)M(n-k),$$ $n\ge 2$.

ⓘ

Symbols:

$k$: nonnegative integer, $n$: nonnegative integer and $M(n)$: Motzkin number

Referenced by:

§26.6(iii)

Permalink:

http://dlmf.nist.gov/26.6.E11

Encodings:

pMML, png, TeX

See also:

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

… ►

26.6.13 $$M(n)=\sum _{k=0}^n{(-1)}^k\left(\genfrac{}{}{0.0pt}{}{n}{k}\right)C\left(n+1-k\right),$$

ⓘ

Symbols:

$C\left(n\right)$: Catalan number, $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, $k$: nonnegative integer, $n$: nonnegative integer and $M(n)$: Motzkin number

Permalink:

http://dlmf.nist.gov/26.6.E13

Encodings:

pMML, png, TeX

See also:

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

…

17: 24.11 Asymptotic Approximations

… ►

24.11.1 $${(-1)}^{n+1}{B}_{2n}\sim \frac{2(2n)!}{{(2\pi )}^{2n}},$$

ⓘ

Symbols:

$B_n$: Bernoulli numbers, $\sim$: asymptotic equality, $\pi$: the ratio of the circumference of a circle to its diameter, $!$: factorial (as in $n!$) and $n$: integer

Referenced by:

§24.11

Permalink:

http://dlmf.nist.gov/24.11.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §24.11 and Ch.24

►

24.11.2 $${(-1)}^{n+1}{B}_{2n}\sim 4\sqrt{\pi n}{\left(\frac{n}{\pi \mathrm{e}}\right)}^{2n},$$

ⓘ

Symbols:

$B_n$: Bernoulli numbers, $\sim$: asymptotic equality, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm and $n$: integer

Permalink:

http://dlmf.nist.gov/24.11.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §24.11 and Ch.24

…

18: 3.11 Approximation Techniques

… ►

3.11.28 $$S=\sum _{j=1}^J{\left(f(x_j)-p_n(x_j)\right)}^2.$$

ⓘ

Symbols:

$p_n(x)$: approximation and $J>n+1$: number of points

Permalink:

http://dlmf.nist.gov/3.11.E28

Encodings:

pMML, png, TeX

See also:

Annotations for §3.11(v), §3.11 and Ch.3

►Here $x_j$, $j=1,2,\mathrm{\dots },J$, is a given set of distinct real points and $J\ge n+1$. … ►

3.11.32 $$\sum _{j=1}^{J}w(x_j){\left(f(x_j)-{\mathrm{\Phi }}_n(x_j)\right)}^2,$$

ⓘ

Symbols:

$J>n+1$: number of points, ${\mathrm{\Phi }}_n(x)$: approximant and $w(x)$: weight function

Referenced by:

§3.11(v)

Permalink:

http://dlmf.nist.gov/3.11.E32

Encodings:

pMML, png, TeX

See also:

Annotations for §3.11(v), §3.11 and Ch.3

… ►

3.11.34 $$X_{k\mathrm{\ell }}=\sum _{j=1}^{J}w(x_j){\varphi }_k(x_j){\varphi }_{\mathrm{\ell }}(x_j),$$

ⓘ

Symbols:

$J>n+1$: number of points, $X_k$: elements, ${\varphi }_k(x)$: functions and $w(x)$: weight function

Permalink:

http://dlmf.nist.gov/3.11.E34

Encodings:

pMML, png, TeX

See also:

Annotations for §3.11(v), §3.11 and Ch.3

… ►

3.11.35 $$F_k=\sum _{j=1}^{J}w(x_j)f(x_j){\varphi }_k(x_j).$$

ⓘ

Symbols:

$J>n+1$: number of points, $F_k$: elements, ${\varphi }_k(x)$: functions and $w(x)$: weight function

Permalink:

http://dlmf.nist.gov/3.11.E35

Encodings:

pMML, png, TeX

See also:

Annotations for §3.11(v), §3.11 and Ch.3

…

19: 24.9 Inequalities

… ►

24.9.6 $$5\sqrt{\pi n}{\left(\frac{n}{\pi \mathrm{e}}\right)}^{2n}>{(-1)}^{n+1}{B}_{2n}>4\sqrt{\pi n}{\left(\frac{n}{\pi \mathrm{e}}\right)}^{2n},$$

ⓘ

Symbols:

$B_n$: Bernoulli numbers, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm and $n$: integer

Referenced by:

§24.9, §24.9

Permalink:

http://dlmf.nist.gov/24.9.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §24.9 and Ch.24

… ►

24.9.8 $$\frac{2(2n)!}{{(2\pi )}^{2n}}\frac{1}{1-2^{\beta -2n}}\ge {(-1)}^{n+1}{B}_{2n}\ge \frac{2(2n)!}{{(2\pi )}^{2n}}\frac{1}{1-2^{-2n}}$$

ⓘ

Symbols:

$B_n$: Bernoulli numbers, $\pi$: the ratio of the circumference of a circle to its diameter, $!$: factorial (as in $n!$), $n$: integer and $\beta$: quantity

Referenced by:

§24.9

Permalink:

http://dlmf.nist.gov/24.9.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §24.9 and Ch.24

… ►

24.9.10 $$\frac{4^{n+1}(2n)!}{{\pi }^{2n+1}}>{(-1)}^n{E}_{2n}>\frac{4^{n+1}(2n)!}{{\pi }^{2n+1}}\frac{1}{1+3^{-1-2n}}.$$

ⓘ

Symbols:

$E_n$: Euler numbers, $\pi$: the ratio of the circumference of a circle to its diameter, $!$: factorial (as in $n!$) and $n$: integer

A&S Ref:

23.1.15

Referenced by:

§24.9

Permalink:

http://dlmf.nist.gov/24.9.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §24.9 and Ch.24

20: 26.7 Set Partitions: Bell Numbers

… ►

26.7.6 $$B\left(n+1\right)=\sum _{k=0}^n\left(\genfrac{}{}{0.0pt}{}{n}{k}\right)B\left(k\right).$$

ⓘ

Symbols:

$B\left(n\right)$: Bell number, $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, $k$: nonnegative integer and $n$: nonnegative integer

Referenced by:

Erratum (V1.0.1) for Equation (26.7.6)

Permalink:

http://dlmf.nist.gov/26.7.E6

Encodings:

pMML, png, TeX

Errata (effective with 1.0.1):

Originally appeared with $B\left(n\right)$ in the summation, instead of $B\left(k\right)$:

$$B\left(n+1\right)=\sum _{k=0}^n\left(\genfrac{}{}{0.0pt}{}{n}{k}\right)B\left(n\right)$$

Reported 2010-11-07 by Layne Watson

See also:

Annotations for §26.7(iii), §26.7 and Ch.26

…