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11: Guide to Searching the DLMF

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a textual word, a number, or a math symbol.

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any double-quoted sequence of textual words and numbers.

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adj, prec/n, and near/n, where n is any positive natural number.

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$	stands for any number of alphanumeric characters
	(the more conventional * is reserved for the multiplication operator)
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12: 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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13: 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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14: 27.2 Functions

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§27.2(i) Definitions

… ►where $p_1,p_2,\mathrm{\dots },p_{\nu \left(n\right)}$ 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 $n$. … ►(See Gauss (1863, Band II, pp. 437–477) and Legendre (1808, p. 394).) … ► … ►

§27.2(ii) Tables

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15: 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: … ►For example, $maj(35247816)=2+6=8$. … ►The Eulerian number, denoted , is the number of permutations in ${𝔖}_n$ with exactly $k$ descents. … ►

§26.14(iii) Identities

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16: 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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17: 27.13 Functions

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§27.13(ii) Goldbach Conjecture

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§27.13(iii) Waring’s Problem

… ►Hence $r_2\left(5\right)=8$ because both divisors, $1$ and $5$, are congruent to $1(mod4)$. … ►By similar methods Jacobi proved that $r_4\left(n\right)=8{\sigma }_1\left(n\right)$ if $n$ is odd, whereas, if $n$ is even, $r_4\left(n\right)=24$ times the sum of the odd divisors of $n$. …Explicit formulas for $r_k\left(n\right)$ have been obtained by similar methods for $k=6,8,10$, and $12$, but they are more complicated. …

18: 1.2 Elementary Algebra

… ►The arithmetic mean of $n$ numbers $a_1,a_2,\mathrm{\dots },a_n$ is … ►The geometric mean $G$ and harmonic mean $H$ of $n$ positive numbers $a_1,a_2,\mathrm{\dots },a_n$ are given by … ►If $r$ is a nonzero real number, then the weighted mean $M(r)$ of $n$ nonnegative numbers $a_1,a_2,\mathrm{\dots },a_n$, and $n$ positive numbers $p_1,p_2,\mathrm{\dots },p_n$ with … ►The dot product notation $𝐮\cdot 𝐯$ is reserved for the physical three-dimensional vectors of (1.6.2). … ►The diagonal elements are not necessarily distinct, and the number of identical (degenerate) diagonal elements is the multiplicity of that specific eigenvalue. …

19: 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. …

20: 27.12 Asymptotic Formulas: Primes

§27.12 Asymptotic Formulas: Primes

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Prime Number Theorem

… ►The number of such primes not exceeding $x$ is … ►There are infinitely many Carmichael numbers.