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1: 24.1 Special Notation

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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: Bibliography M

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Computational Algebra Group, School of Mathematics and Statistics, University of Sydney, Australia.

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Notes:

A software package (the successor to CAYLEY) designed to solve computationally hard problems in algebra, number theory, geometry, and combinatorics, and to provide a mathematically rigorous environment for computing with algebraic, number-theoretic, combinatoric, and geometric objects.

External Links:

Magma

Referenced by:

4th item.

Encodings:

BibTeX

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Exceptional orthogonal polynomials.

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Notes:

Lecture at ICMAT School on Orthogonal Polynomials in Approximation Theory and Mathematical Physics, available at https://www.icmat.es/RT/optrim/school/lectures/Milson/icmat17.pdf

Referenced by:

§18.36(vi), §18.38(iii).

Encodings:

BibTeX

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On the representation of numbers as a sum of $2r$ squares. Quarterly Journal of Math. 48, pp. 93–104.

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External Links:

ZentralBlatt

Referenced by:

§27.13(iv).

Encodings:

BibTeX

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Asymptotic development of the Stirling numbers of the first kind. J. London Math. Soc. 33, pp. 133–146.

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External Links:

Document, MathReview (N. J. Fine), ZentralBlatt

Referenced by:

§26.1, §26.8(vii).

Encodings:

BibTeX

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Stirling numbers of the second kind. Duke Math. J. 25 (1), pp. 29–43.

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External Links:

Document, MathReview (A. Sade), ZentralBlatt

Referenced by:

§26.1, §26.8(vii), Notations S.

Encodings:

BibTeX

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3: 27.18 Methods of Computation: Primes

§27.18 Methods of Computation: Primes

►An overview of methods for precise counting of the number of primes not exceeding an arbitrary integer $x$ is given in Crandall and Pomerance (2005, §3.7). …An analytic approach using a contour integral of the Riemann zeta function (§25.2(i)) is discussed in Borwein et al. (2000). … ►These algorithms are used for testing primality of Mersenne numbers, $2^n-1$, and Fermat numbers, $2^{2^n}+1$. …

4: 26.11 Integer Partitions: Compositions

… ► $c\left(n\right)$ 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 $T=\{2,3,4,\mathrm{\dots }\}$. … ►

26.11.1 $$c\left(0\right)=c(\in T,0)=1.$$

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Symbols:

$\in$: element of, $c\left(n\right)$: number of compositions of $n$ , $c(\mathrm{condition},n)$: restricted number of compositions of $n$ and $T$: set

Permalink:

http://dlmf.nist.gov/26.11.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §26.11 and Ch.26

… ►The Fibonacci numbers are determined recursively by … ►Additional information on Fibonacci numbers can be found in Rosen et al. (2000, pp. 140–145).

5: 26.6 Other Lattice Path Numbers

§26.6 Other Lattice Path Numbers

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Delannoy Number $D(m,n)$

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Motzkin Number $M(n)$

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Narayana Number $N(n,k)$

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§26.6(iv) Identities

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6: 24.15 Related Sequences of Numbers

§24.15 Related Sequences of Numbers

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§24.15(i) Genocchi Numbers

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§24.15(ii) Tangent Numbers

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§24.15(iii) Stirling Numbers

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§24.15(iv) Fibonacci and Lucas Numbers

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7: 26.5 Lattice Paths: Catalan Numbers

§26.5 Lattice Paths: Catalan Numbers

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§26.5(i) Definitions

► $C\left(n\right)$ is the Catalan number. … ►

§26.5(ii) Generating Function

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§26.5(iii) Recurrence Relations

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8: 26.14 Permutations: Order Notation

… ►As an example, $35247816$ is an element of ${𝔖}_8.$ The inversion number is the number of pairs of elements for which the larger element precedes the smaller: … ► ►The Eulerian number, denoted , is the number of permutations in ${𝔖}_n$ with exactly $k$ descents. …The Eulerian number is equal to the number of permutations in ${𝔖}_n$ with exactly $k$ excedances. … ►

§26.14(iii) Identities

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9: 26.7 Set Partitions: Bell Numbers

§26.7 Set Partitions: Bell Numbers

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§26.7(i) Definitions

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§26.7(ii) Generating Function

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§26.7(iii) Recurrence Relation

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§26.7(iv) Asymptotic Approximation

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10: 26.8 Set Partitions: Stirling Numbers

§26.8 Set Partitions: Stirling Numbers

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§26.8(i) Definitions

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§26.8(v) Identities

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§26.8(vi) Relations to Bernoulli Numbers

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