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21: Bruce R. Miller

… ►While developing the supporting theories, he discovered a passion for symbolic computation and computer algebra. …

22: Jim Pitman

… ►org and to provide technical support to other organizations willing to do the same. …

23: 24.15 Related Sequences of Numbers

§24.15 Related Sequences of Numbers

►

§24.15(i) Genocchi Numbers

… ►

§24.15(ii) Tangent Numbers

… ►

§24.15(iii) Stirling Numbers

… ►

§24.15(iv) Fibonacci and Lucas Numbers

…

24: 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

…

25: 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

…

26: 26.7 Set Partitions: Bell Numbers

§26.7 Set Partitions: Bell Numbers

►

§26.7(i) Definitions

… ►

§26.7(ii) Generating Function

… ►

§26.7(iii) Recurrence Relation

… ►

§26.7(iv) Asymptotic Approximation

…

27: 20.12 Mathematical Applications

… ►

§20.12(i) Number Theory

►For applications of

\theta_{3}\left(0,q\right)

to problems involving sums of squares of integers see §27.13(iv), and for extensions see Estermann (1959), Serre (1973, pp. 106–109), Koblitz (1993, pp. 176–177), and McKean and Moll (1999, pp. 142–143). ►For applications of Jacobi’s triple product (20.5.9) to Ramanujan’s

\tau\left(n\right)

function and Euler’s pentagonal numbers see Hardy and Wright (1979, pp. 132–160) and McKean and Moll (1999, pp. 143–145). … ►For the terminology and notation see McKean and Moll (1999, pp. 48–53). ►The space of complex tori

\mathbb{C}/(\mathbb{Z}+\tau\mathbb{Z})

(that is, the set of complex numbers

z

in which two of these numbers

z_{1}

and

z_{2}

are regarded as equivalent if there exist integers

m, n

such that

z_{1}-z_{2}=m+\tau n

) is mapped into the projective space

P^{3}

via the identification

z\to(\theta_{1}\left(2z\middle|\tau\right),\theta_{2}\left(2z\middle|\tau% \right),\theta_{3}\left(2z\middle|\tau\right),\theta_{4}\left(2z\middle|\tau% \right))

. …

28: 26.8 Set Partitions: Stirling Numbers

§26.8 Set Partitions: Stirling Numbers

►

§26.8(i) Definitions

… ► … ►

§26.8(v) Identities

… ►

§26.8(vi) Relations to Bernoulli Numbers

…

29: 24.19 Methods of Computation

… ►

§24.19(i) Bernoulli and Euler Numbers and Polynomials

►Equations (24.5.3) and (24.5.4) enable

B_{n}

and

E_{n}

to be computed by recurrence. …A similar method can be used for the Euler numbers based on (4.19.5). … ►

§24.19(ii) Values of $B_{n}$ Modulo $p$

… ►We list here three methods, arranged in increasing order of efficiency. …

30: 27.17 Other Applications

§27.17 Other Applications

►Reed et al. (1990, pp. 458–470) describes a number-theoretic approach to Fourier analysis (called the arithmetic Fourier transform) that uses the Möbius inversion (27.5.7) to increase efficiency in computing coefficients of Fourier series. ►Congruences are used in constructing perpetual calendars, splicing telephone cables, scheduling round-robin tournaments, devising systematic methods for storing computer files, and generating pseudorandom numbers. … ►There are also applications of number theory in many diverse areas, including physics, biology, chemistry, communications, and art. …