Catalan%20numbers

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1: 24.1 Special Notation

… ►

Bernoulli Numbers and Polynomials

►The origin of the notation

B_{n}

B_{n}\left(x\right)

, is not clear. … ►

Euler Numbers and Polynomials

… ►Its coefficients were first studied in Euler (1755); they were called Euler numbers by Raabe in 1851. The notations

E_{n}

E_{n}\left(x\right)

, as defined in §24.2(ii), were used in Lucas (1891) and Nörlund (1924). …

2: 26.5 Lattice Paths: Catalan Numbers

§26.5 Lattice Paths: Catalan Numbers

►

§26.5(i) Definitions

►

C\left(n\right)

is the Catalan number. … ►

§26.5(ii) Generating Function

… ►

§26.5(iii) Recurrence Relations

…

3: 26.6 Other Lattice Path Numbers

§26.6 Other Lattice Path Numbers

… ►

Delannoy Number $D(m,n)$

… ►

Motzkin Number $M(n)$

… ►

Narayana Number $N(n,k)$

… ►

§26.6(iv) Identities

…

4: 26.1 Special Notation

… ► ►►►►►

$\genfrac{(}{)}{0.0pt}{}{m}{n}$	binomial coefficient.
…
$\genfrac{<}{>}{0.0pt}{}{m}{n}$	Eulerian number.
…
$B\left(n\right)$	Bell number.
$C\left(n\right)$	Catalan number.
…

… ►Other notations for

s\left(n,k\right)

, the Stirling numbers of the first kind, include

S_{n}^{(k)}

(Abramowitz and Stegun (1964, Chapter 24), Fort (1948)),

S_{n}^{k}

(Jordan (1939), Moser and Wyman (1958a)),

\genfrac{(}{)}{0.0pt}{}{n-1}{k-1}B_{n-k}^{(n)}

(Milne-Thomson (1933)),

(-1)^{n-k}S_{1}(n-1,n-k)

(Carlitz (1960), Gould (1960)),

(-1)^{n-k}\left[n\atop k\right]

(Knuth (1992), Graham et al. (1994), Rosen et al. (2000)). ►Other notations for

S\left(n,k\right)

, the Stirling numbers of the second kind, include

\mathscr{S}^{(k)}_{n}

(Fort (1948)),

\mathfrak{S}_{n}^{k}

(Jordan (1939)),

\sigma_{n}^{k}

(Moser and Wyman (1958b)),

\genfrac{(}{)}{0.0pt}{}{n}{k}B_{n-k}^{(-k)}

(Milne-Thomson (1933)),

S_{2}(k,n-k)

(Carlitz (1960), Gould (1960)),

\left\{n\atop k\right\}

(Knuth (1992), Graham et al. (1994), Rosen et al. (2000)), and also an unconventional symbol in Abramowitz and Stegun (1964, Chapter 24).

5: 25.11 Hurwitz Zeta Function

… ►

See accompanying text — Figure 25.11.1: Hurwitz zeta function $\zeta\left(x,a\right)$ , $a$ = 0. …8, 1, $-20\leq x\leq 10$ . … Magnify

… ►where

H_{n}

are the harmonic numbers: … ►

25.11.39

\sum_{k=2}^{\infty}\frac{k}{2^{k}}\zeta\left(k+1,\tfrac{3}{4}\right)=8G,

ⓘ

Symbols:: $\zeta\left(\NVar{s},\NVar{a}\right)$ : Hurwitz zeta function, $k$ : nonnegative integer and $G$ : Catalan’s constant
Keywords:: Catalan’s constant, infinite series
Source:: Adamchik and Srivastava (1998, (2.30), p. 138)
Permalink:: http://dlmf.nist.gov/25.11.E39
Encodings:: pMML, png, TeX
See also:: Annotations for §25.11(xi), §25.11 and Ch.25

►where

G

is Catalan’s constant: ►

25.11.40

G\equiv\sum_{n=0}^{\infty}\frac{(-1)^{n}}{(2n+1)^{2}}=0.91596\;55941\;772\dots.

ⓘ

Symbols:: $\equiv$ : equals by definition, $n$ : nonnegative integer and $G$ : Catalan’s constant
Keywords:: Catalan’s constant, definition
Source:: Adamchik and Srivastava (1998, (2.32), p. 139)
Notes:: For more digits see OEIS Sequence A006752; see also Sloane (2003).
Permalink:: http://dlmf.nist.gov/25.11.E40
Encodings:: pMML, png, TeX
See also:: Annotations for §25.11(xi), §25.11 and Ch.25

…

6: 27.15 Chinese Remainder Theorem

§27.15 Chinese Remainder Theorem

… ►This theorem is employed to increase efficiency in calculating with large numbers by making use of smaller numbers in most of the calculation. …Their product

m

has 20 digits, twice the number of digits in the data. …These numbers, in turn, are combined by the Chinese remainder theorem to obtain the final result

\pmod{m}

, which is correct to 20 digits. …

7: 27.2 Functions

… ►where

p_{1},p_{2},\dots,p_{\nu\left(n\right)}

are the distinct prime factors of

n

, each exponent

a_{r}

is positive, and

\nu\left(n\right)

is the number of distinct primes dividing

n

. …Euclid’s Elements (Euclid (1908, Book IX, Proposition 20)) gives an elegant proof that there are infinitely many primes. … ►(See Gauss (1863, Band II, pp. 437–477) and Legendre (1808, p. 394).) … ► … ►

§27.2(ii) Tables

…

8: 24.20 Tables

§24.20 Tables

… ►Wagstaff (1978) gives complete prime factorizations of

N_{n}

and

E_{n}

for

n=20(2)60

and

n=8(2)42

, respectively. …

9: 26.14 Permutations: Order Notation

… ►As an example,

35247816

is an element of

\mathfrak{S}_{8}.

The inversion number is the number of pairs of elements for which the larger element precedes the smaller: … ► ►The Eulerian number, denoted

\genfrac{<}{>}{0.0pt}{}{n}{k}

, is the number of permutations in

\mathfrak{S}_{n}

with exactly

k

descents. …The Eulerian number

\genfrac{<}{>}{0.0pt}{}{n}{k}

is equal to the number of permutations in

\mathfrak{S}_{n}

with exactly

k

excedances. … ►

§26.14(iii) Identities

…

10: 26.13 Permutations: Cycle Notation

… ►The Stirling cycle numbers of the first kind, denoted by

\genfrac{[}{]}{0.0pt}{}{n}{k}

, count the number of permutations of

\{1,2,\ldots,n\}

with exactly

k

cycles. They are related to Stirling numbers of the first kind by …See §26.8 for generating functions, recurrence relations, identities, and asymptotic approximations. … ►The derangement number,

d(n)

, is the number of elements of

\mathfrak{S}_{n}

with no fixed points: … ►A permutation is even or odd according to the parity of the number of transpositions. …

Catalan%20numbers

1—10 of 274 matching pages

1: 24.1 Special Notation

Bernoulli Numbers and Polynomials

Euler Numbers and Polynomials

2: 26.5 Lattice Paths: Catalan Numbers

§26.5 Lattice Paths: Catalan Numbers

§26.5(i) Definitions

§26.5(ii) Generating Function

§26.5(iii) Recurrence Relations

3: 26.6 Other Lattice Path Numbers

§26.6 Other Lattice Path Numbers

Delannoy Number D ⁡ ( m , n )

Motzkin Number M ⁡ ( n )

Narayana Number N ⁡ ( n , k )

§26.6(iv) Identities

4: 26.1 Special Notation

5: 25.11 Hurwitz Zeta Function

6: 27.15 Chinese Remainder Theorem

§27.15 Chinese Remainder Theorem

7: 27.2 Functions

§27.2(ii) Tables

8: 24.20 Tables

§24.20 Tables

9: 26.14 Permutations: Order Notation

§26.14(iii) Identities

10: 26.13 Permutations: Cycle Notation

Delannoy Number $D(m,n)$

Motzkin Number $M(n)$

Narayana Number $N(n,k)$