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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: 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).

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: 26.6 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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5: 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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6: 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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7: 25.11 Hurwitz Zeta Function

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25.11.22 $${\zeta }^{\prime }(1-2n,\frac{1}{2})=-\frac{B_{2n}\ln 2}{n\cdot 4^n}-\frac{(2^{2n-1}-1){\zeta }^{\prime }\left(1-2n\right)}{2^{2n-1}},$$ $n=1,2,3,\mathrm{\dots }$.

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Symbols:

$B_n$: Bernoulli numbers, $\zeta (s,a)$: Hurwitz zeta function, $\zeta \left(s\right)$: Riemann zeta function, $\ln z$: principal branch of logarithm function and $n$: nonnegative integer

Keywords:

derivative

Source:

Miller and Adamchik (1998, (17), p. 205)

Permalink:

http://dlmf.nist.gov/25.11.E22

Encodings:

pMML, png, TeX

See also:

Annotations for §25.11(vi), §25.11(vi), §25.11 and Ch.25

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25.11.23 $${\zeta }^{\prime }(1-2n,\frac{1}{3})=-\frac{\pi (9^n-1)B_{2n}}{8n\sqrt{3}(3^{2n-1}-1)}-\frac{B_{2n}\ln 3}{4n\cdot 3^{2n-1}}-\frac{{(-1)}^n{\psi }^{(2n-1)}\left(\frac{1}{3}\right)}{2\sqrt{3}{(6\pi )}^{2n-1}}-\frac{\left(3^{2n-1}-1\right){\zeta }^{\prime }\left(1-2n\right)}{2\cdot 3^{2n-1}},$$ $n=1,2,3,\mathrm{\dots }$.

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Symbols:

$B_n$: Bernoulli numbers, $\zeta (s,a)$: Hurwitz zeta function, $\zeta \left(s\right)$: Riemann zeta function, $\pi$: the ratio of the circumference of a circle to its diameter, $\psi \left(z\right)$: psi (or digamma) function, $\ln z$: principal branch of logarithm function and $n$: nonnegative integer

Keywords:

derivative

Source:

Miller and Adamchik (1998, (18), p. 205)

Permalink:

http://dlmf.nist.gov/25.11.E23

Encodings:

pMML, png, TeX

See also:

Annotations for §25.11(vi), §25.11(vi), §25.11 and Ch.25

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25.11.32 $${\int }_0^a{x}^n\psi \left(x\right)dx={(-1)}^{n-1}{\zeta }^{\prime }\left(-n\right)+{(-1)}^n{H}_n\frac{B_{n+1}}{n+1}-\sum _{k=0}^n{(-1)}^k\left(\genfrac{}{}{0.0pt}{}{n}{k}\right)H_k\frac{B_{k+1}(a)}{k+1}{a}^{n-k}+\sum _{k=0}^n{(-1)}^k\left(\genfrac{}{}{0.0pt}{}{n}{k}\right){\zeta }^{\prime }(-k,a)a^{n-k},$$ $n=1,2,\mathrm{\dots }$, $\mathrm{\Re }a>0$,

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Symbols:

$B_n$: Bernoulli numbers, $\zeta (s,a)$: Hurwitz zeta function, $\zeta \left(s\right)$: Riemann zeta function, $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, $dx$: differential of $x$ , $\psi \left(z\right)$: psi (or digamma) function, $H_n$: harmonic number, $\int$: integral, $\mathrm{\Re }$: real part, $k$: nonnegative integer, $n$: nonnegative integer, $x$: real variable, $a$: real or complex parameter and $h(n)$: harmonic number

Keywords:

definite integral, integral representation

Source:

Adamchik (1998, (17), p. 197)

Referenced by:

Erratum (V1.1.4) for Notation

Permalink:

http://dlmf.nist.gov/25.11.E32

Encodings:

pMML, png, TeX

Notation (effective with 1.1.4):

The notation previously used for the harmonic number $h(n)$ (resp. $h(k)$) has been replaced to be $H_n$ (resp. $H_k$).

Suggested 2021-08-23 by Gergő Nemes

See also:

Annotations for §25.11(viii), §25.11 and Ch.25

►where $H_n$ are the harmonic numbers: ►

25.11.33 $$H_n=\sum _{k=1}^n{k}^{-1}.$$

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Defines:

$H_n$: harmonic number

Symbols:

$k$: nonnegative integer, $n$: nonnegative integer and $h(n)$: harmonic number

Keywords:

harmonic number

Source:

Knuth (1968, Section 1.2.7, p. 73)

Referenced by:

§25.11(viii), §25.16(ii), Erratum (V1.1.4) for Notation

Permalink:

http://dlmf.nist.gov/25.11.E33

Encodings:

pMML, png, TeX

Notation (effective with 1.1.4):

The notation previously used for the harmonic number $h(n)$ has been replaced to be $H_n$.

Suggested 2021-08-23 by Gergő Nemes

See also:

Annotations for §25.11(viii), §25.11 and Ch.25

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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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