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Handbook of mathematical functions with formulas, graphs and mathematical tables. (English) Zbl 0171.38503

Washington: U.S. Department of Commerce. xiv, 1046 pp. (1964).
Dieses umfassende Werk über das Gebiet der speziellen Funktionen vereint eine Vielzahl von Tafeln und dazugehörigen Formeln. 29 Kapitel wurden von 28 Autoren bearbeitet. Die Tafeln sind teilweise von sehr hoher Genauigkeit, z. B. sind die trigonometrischen Funktionen mit 23 Stellen wiedergegeben. Im einzelnen sind in dem Buch Tafeln enthalten über mathematische und physikalische Konstanten, elementare transzendente Funktionen, Integralsinus und verwandte Funktionen, Gammafunktionen und Verwandte, Fehlerintegral und Fresnelsche Integrale, Legendresche Funktionen, Besselsche Funktionen und Integrale, Struvesche Funktionen und Verwandte, hypergeometrische und konfluente hypergeometrische Funktionen, elliptische Funktionen und Integrale, parabolische Zylinderfunktionen und eine Anzahl weiterer spezieller Funktionen.
Ein Kapitel unter der Überschrift ,, Elementare analytische Methoden” enthält eine nützliche Formelsammlung und Tafeln von Potenzen und Wurzeln. Ein weiteres Kapitel ist der numerischen Integration, Differentiation und Interpolation gewidmet und enthält ebenfalls eine Anzahl von Tafeln, etwa die Lagrangeschen Interpolationskoeffizienten bis achter Ordnung oder Abzissen und Gewichte der Gaußschen Quadraturformeln auf 20 Stellen. In weiteren Kapiteln werden Mathieusche Funktionen, Orthogonalpolynome, Bernoullische und Eulersche Polynome sowie die Riemannsche Zetafunktion, statistische Verteilungsfunktion und Laplace-Transformationen behandelt.
Ein umfangreiches Kapitel ist der Kombinatorik und zahlentheoretischen Funktionen gewidmet.
Mit diesem Buch dürfte das Standardtafelwerk vorliegen, das für viele Zwecke spezielle und umfangreiche Tafeln und Formelsammlungen erasetzen kann oder sogar übertrifft.
Table Errata, see Math. Comput. 21, 747 (1967).
Reviewer: K.-H. Bachmann

MSC:

33-00 General reference works (handbooks, dictionaries, bibliographies, etc.) pertaining to special functions
00A20 Dictionaries and other general reference works
00A22 Formularies
65A05 Tables in numerical analysis
65Dxx Numerical approximation and computational geometry (primarily algorithms)
41A55 Approximate quadratures
62Q05 Statistical tables
44A10 Laplace transform
11B68 Bernoulli and Euler numbers and polynomials
11M06 \(\zeta (s)\) and \(L(s, \chi)\)
11Y70 Values of arithmetic functions; tables

Digital Library of Mathematical Functions:

§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions
§10.21(ix) Complex Zeros ‣ §10.21 Zeros ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions
§10.47(i) Differential Equations ‣ §10.47 Definitions and Basic Properties ‣ Spherical Bessel Functions ‣ Chapter 10 Bessel Functions
§10.68(iv) Further Properties ‣ §10.68 Modulus and Phase Functions ‣ Kelvin Functions ‣ Chapter 10 Bessel Functions
§10.70 Zeros ‣ Kelvin Functions ‣ Chapter 10 Bessel Functions
2nd item ‣ §10.75(xi) Kelvin Functions and their Derivatives ‣ §10.75 Tables ‣ Computation ‣ Chapter 10 Bessel Functions
4th item ‣ §10.75(ii) Bessel Functions and their Derivatives ‣ §10.75 Tables ‣ Computation ‣ Chapter 10 Bessel Functions
4th item ‣ Real Zeros ‣ §10.75(iii) Zeros and Associated Values of the Bessel Functions, Hankel Functions, and their Derivatives ‣ §10.75 Tables ‣ Computation ‣ Chapter 10 Bessel Functions
5th item ‣ Real Zeros ‣ §10.75(iii) Zeros and Associated Values of the Bessel Functions, Hankel Functions, and their Derivatives ‣ §10.75 Tables ‣ Computation ‣ Chapter 10 Bessel Functions
1st item ‣ Complex Zeros ‣ §10.75(iii) Zeros and Associated Values of the Bessel Functions, Hankel Functions, and their Derivatives ‣ §10.75 Tables ‣ Computation ‣ Chapter 10 Bessel Functions
1st item ‣ §10.75(iv) Integrals of Bessel Functions ‣ §10.75 Tables ‣ Computation ‣ Chapter 10 Bessel Functions
4th item ‣ §10.75(v) Modified Bessel Functions and their Derivatives ‣ §10.75 Tables ‣ Computation ‣ Chapter 10 Bessel Functions
1st item ‣ §10.75(vii) Integrals of Modified Bessel Functions ‣ §10.75 Tables ‣ Computation ‣ Chapter 10 Bessel Functions
1st item ‣ §10.75(ix) Spherical Bessel Functions, Modified Spherical Bessel Functions, and their Derivatives ‣ §10.75 Tables ‣ Computation ‣ Chapter 10 Bessel Functions
Chapter 10 Bessel Functions
§11.10(vi) Relations to Other Functions ‣ §11.10 Anger–Weber Functions ‣ Related Functions ‣ Chapter 11 Struve and Related Functions
1st item ‣ §11.14(ii) Struve Functions ‣ §11.14 Tables ‣ Computation ‣ Chapter 11 Struve and Related Functions
1st item ‣ §11.14(iii) Integrals ‣ §11.14 Tables ‣ Computation ‣ Chapter 11 Struve and Related Functions
Chapter 11 Struve and Related Functions
1st item ‣ §12.19 Tables ‣ Computation ‣ Chapter 12 Parabolic Cylinder Functions
Chapter 12 Parabolic Cylinder Functions
3rd item ‣ §13.30 Tables ‣ Computation ‣ Chapter 13 Confluent Hypergeometric Functions
Chapter 13 Confluent Hypergeometric Functions
1st item ‣ §14.33 Tables ‣ Computation ‣ Chapter 14 Legendre and Related Functions
Chapter 14 Legendre and Related Functions
Chapter 15 Hypergeometric Function
Hermite ‣ §18.14(i) Upper Bounds ‣ §18.14 Inequalities ‣ Classical Orthogonal Polynomials ‣ Chapter 18 Orthogonal Polynomials
§18.3 Definitions ‣ Classical Orthogonal Polynomials ‣ Chapter 18 Orthogonal Polynomials
§18.41(ii) Zeros ‣ §18.41 Tables ‣ Computation ‣ Chapter 18 Orthogonal Polynomials
§18.41(i) Polynomials ‣ §18.41 Tables ‣ Computation ‣ Chapter 18 Orthogonal Polynomials
Hermite ‣ §18.5(iv) Numerical Coefficients ‣ §18.5 Explicit Representations ‣ Classical Orthogonal Polynomials ‣ Chapter 18 Orthogonal Polynomials
§18.8 Differential Equations ‣ Classical Orthogonal Polynomials ‣ Chapter 18 Orthogonal Polynomials
§19.14(ii) General Case ‣ §19.14 Reduction of General Elliptic Integrals ‣ Legendre’s Integrals ‣ Chapter 19 Elliptic Integrals
§19.14(i) Examples ‣ §19.14 Reduction of General Elliptic Integrals ‣ Legendre’s Integrals ‣ Chapter 19 Elliptic Integrals
§19.1 Special Notation ‣ Notation ‣ Chapter 19 Elliptic Integrals
§19.25(v) Jacobian Elliptic Functions ‣ §19.25 Relations to Other Functions ‣ Symmetric Integrals ‣ Chapter 19 Elliptic Integrals
§19.36(iii) Via Theta Functions ‣ §19.36 Methods of Computation ‣ Computation ‣ Chapter 19 Elliptic Integrals
Functions K ( k ) and E ( k ) ‣ §19.37(ii) Legendre’s Complete Integrals ‣ §19.37 Tables ‣ Computation ‣ Chapter 19 Elliptic Integrals
Functions K ( k ) and E ( k ) ‣ §19.37(ii) Legendre’s Complete Integrals ‣ §19.37 Tables ‣ Computation ‣ Chapter 19 Elliptic Integrals
Function exp ( - / ⁢ π K ′ ( k ) K ( k ) ) ( = q ( k ) ) ‣ §19.37(ii) Legendre’s Complete Integrals ‣ §19.37 Tables ‣ Computation ‣ Chapter 19 Elliptic Integrals
Function exp ( - / ⁢ π K ′ ( k ) K ( k ) ) ( = q ( k ) ) ‣ §19.37(ii) Legendre’s Complete Integrals ‣ §19.37 Tables ‣ Computation ‣ Chapter 19 Elliptic Integrals
Functions F ( ϕ , k ) and E ( ϕ , k ) ‣ §19.37(iii) Legendre’s Incomplete Integrals ‣ §19.37 Tables ‣ Computation ‣ Chapter 19 Elliptic Integrals
Function Π ( ϕ , α 2 , k ) ‣ §19.37(iii) Legendre’s Incomplete Integrals ‣ §19.37 Tables ‣ Computation ‣ Chapter 19 Elliptic Integrals
§19.38 Approximations ‣ Computation ‣ Chapter 19 Elliptic Integrals
Chapter 19 Elliptic Integrals
§20.15 Tables ‣ Computation ‣ Chapter 20 Theta Functions
Chapter 20 Theta Functions
§22.15(ii) Representations as Elliptic Integrals ‣ §22.15 Inverse Functions ‣ Properties ‣ Chapter 22 Jacobian Elliptic Functions
§22.1 Special Notation ‣ Notation ‣ Chapter 22 Jacobian Elliptic Functions
Chapter 22 Jacobian Elliptic Functions
Other Notations ‣ §23.1 Special Notation ‣ Notation ‣ Chapter 23 Weierstrass Elliptic and Modular Functions
Rhombic Lattice ‣ §23.20(i) Conformal Mappings ‣ §23.20 Mathematical Applications ‣ Applications ‣ Chapter 23 Weierstrass Elliptic and Modular Functions
§23.23 Tables ‣ Computation ‣ Chapter 23 Weierstrass Elliptic and Modular Functions
§23.23 Tables ‣ Computation ‣ Chapter 23 Weierstrass Elliptic and Modular Functions
§23.9 Laurent and Other Power Series ‣ Weierstrass Elliptic Functions ‣ Chapter 23 Weierstrass Elliptic and Modular Functions
§23.9 Laurent and Other Power Series ‣ Weierstrass Elliptic Functions ‣ Chapter 23 Weierstrass Elliptic and Modular Functions
§23.9 Laurent and Other Power Series ‣ Weierstrass Elliptic Functions ‣ Chapter 23 Weierstrass Elliptic and Modular Functions
Chapter 23 Weierstrass Elliptic and Modular Functions
§24.20 Tables ‣ Computation ‣ Chapter 24 Bernoulli and Euler Polynomials
§24.2(iv) Tables ‣ §24.2 Definitions and Generating Functions ‣ Properties ‣ Chapter 24 Bernoulli and Euler Polynomials
Chapter 24 Bernoulli and Euler Polynomials
1st item ‣ §25.19 Tables ‣ Computation ‣ Chapter 25 Zeta and Related Functions
Alternative Notations ‣ §26.1 Special Notation ‣ Notation ‣ Chapter 26 Combinatorial Analysis
Alternative Notations ‣ §26.1 Special Notation ‣ Notation ‣ Chapter 26 Combinatorial Analysis
§26.21 Tables ‣ Computation ‣ Chapter 26 Combinatorial Analysis
§26.2 Basic Definitions ‣ Properties ‣ Chapter 26 Combinatorial Analysis
§26.3(i) Definitions ‣ §26.3 Lattice Paths: Binomial Coefficients ‣ Properties ‣ Chapter 26 Combinatorial Analysis
§26.4(i) Definitions ‣ §26.4 Lattice Paths: Multinomial Coefficients and Set Partitions ‣ Properties ‣ Chapter 26 Combinatorial Analysis
§26.8(i) Definitions ‣ §26.8 Set Partitions: Stirling Numbers ‣ Properties ‣ Chapter 26 Combinatorial Analysis
§27.21 Tables ‣ Computation ‣ Chapter 27 Functions of Number Theory
§27.2(ii) Tables ‣ §27.2 Functions ‣ Multiplicative Number Theory ‣ Chapter 27 Functions of Number Theory
Abramowitz and Stegun (1964, Chapter 20) ‣ §28.1 Special Notation ‣ Notation ‣ Chapter 28 Mathieu Functions and Hill’s Equation
Chapter 28 Mathieu Functions and Hill’s Equation
Other Notations ‣ §30.1 Special Notation ‣ Notation ‣ Chapter 30 Spheroidal Wave Functions
Chapter 30 Spheroidal Wave Functions
1st item ‣ §33.24 Tables ‣ Computation ‣ Chapter 33 Coulomb Functions
Chapter 33 Coulomb Functions
§4.44 Other Applications ‣ Applications ‣ Chapter 4 Elementary Functions
§4.46 Tables ‣ Computation ‣ Chapter 4 Elementary Functions
Terminology ‣ §5.11(i) Poincaré-Type Expansions ‣ §5.11 Asymptotic Expansions ‣ Properties ‣ Chapter 5 Gamma Function
§5.22(iii) Complex Variables ‣ §5.22 Tables ‣ Computation ‣ Chapter 5 Gamma Function
§5.22(ii) Real Variables ‣ §5.22 Tables ‣ Computation ‣ Chapter 5 Gamma Function
§5.7(i) Maclaurin and Taylor Series ‣ §5.7 Series Expansions ‣ Properties ‣ Chapter 5 Gamma Function
Chapter 5 Gamma Function
1st item ‣ §6.19(ii) Real Variables ‣ §6.19 Tables ‣ Computation ‣ Chapter 6 Exponential, Logarithmic, Sine, and Cosine Integrals
1st item ‣ §6.19(iii) Complex Variables, = z + x ⁢ i y ‣ §6.19 Tables ‣ Computation ‣ Chapter 6 Exponential, Logarithmic, Sine, and Cosine Integrals
1st item ‣ §6.20(i) Approximations in Terms of Elementary Functions ‣ §6.20 Approximations ‣ Computation ‣ Chapter 6 Exponential, Logarithmic, Sine, and Cosine Integrals
Chapter 6 Exponential, Logarithmic, Sine, and Cosine Integrals
1st item ‣ §7.23(ii) Real Variables ‣ §7.23 Tables ‣ Computation ‣ Chapter 7 Error Functions, Dawson’s and Fresnel Integrals
2nd item ‣ §7.23(ii) Real Variables ‣ §7.23 Tables ‣ Computation ‣ Chapter 7 Error Functions, Dawson’s and Fresnel Integrals
1st item ‣ §7.23(iii) Complex Variables, = z + x ⁢ i y ‣ §7.23 Tables ‣ Computation ‣ Chapter 7 Error Functions, Dawson’s and Fresnel Integrals
Chapter 7 Error Functions, Dawson’s and Fresnel Integrals
(8.17.24) ‣ §8.17(vii) Addendum to 8.17(i) Definitions and Basic Properties ‣ §8.17 Incomplete Beta Functions ‣ Related Functions ‣ Chapter 8 Incomplete Gamma and Related Functions
§8.22(ii) Riemann Zeta Function and Incomplete Riemann Zeta Function ‣ §8.22 Mathematical Applications ‣ Applications ‣ Chapter 8 Incomplete Gamma and Related Functions
1st item ‣ §8.26(iv) Generalized Exponential Integral ‣ §8.26 Tables ‣ Computation ‣ Chapter 8 Incomplete Gamma and Related Functions
Chapter 8 Incomplete Gamma and Related Functions
1st item ‣ §9.18(ii) Real Variables ‣ §9.18 Tables ‣ Computation ‣ Chapter 9 Airy and Related Functions
1st item ‣ §9.18(iv) Zeros ‣ §9.18 Tables ‣ Computation ‣ Chapter 9 Airy and Related Functions
1st item ‣ §9.18(v) Integrals ‣ §9.18 Tables ‣ Computation ‣ Chapter 9 Airy and Related Functions
§9.8(i) Definitions ‣ §9.8 Modulus and Phase ‣ Airy Functions ‣ Chapter 9 Airy and Related Functions
Chapter 9 Airy and Related Functions
Profile Frank W. J. Olver ‣ About the Project
About the Project
Preface ‣ About the Project
Possible Errors in DLMF ‣ Need Help?
Notations E ‣ Notations
Notations E ‣ Notations
Notations F ‣ Notations
Notations F ‣ Notations
Notations H ‣ Notations
Notations H ‣ Notations
Notations J ‣ Notations
Notations K ‣ Notations
Notations P ‣ Notations
Notations P ‣ Notations
Notations S ‣ Notations
Notations Y ‣ Notations

Online Encyclopedia of Integer Sequences:

Expansion of Product_{m >= 1} (1 + x^m); number of partitions of n into distinct parts; number of partitions of n into odd parts.
Euler totient function phi(n): count numbers <= n and prime to n.
Powers of 2: a(n) = 2^n.
Triangular numbers: a(n) = binomial(n+1,2) = n*(n+1)/2 = 0 + 1 + 2 + ... + n.
Schroeder’s second problem (generalized parentheses); also called super-Catalan numbers or little Schroeder numbers.
Sum of proper divisors (or aliquot parts) of n: sum of divisors of n that are less than n.
Primes == +-1 (mod 8).
Double factorial of odd numbers: a(n) = (2*n-1)!! = 1*3*5*...*(2*n-1).
sigma_3(n): sum of cubes of divisors of n.
sigma_4(n): sum of 4th powers of divisors of n.
sigma_5(n), the sum of the 5th powers of the divisors of n.
Stirling’s formula: numerators of asymptotic series for Gamma function.
Number of prime divisors of n counted with multiplicity (also called big omega of n, bigomega(n) or Omega(n)).
Lesser of twin primes.
a(n) = binomial(n, floor(n/2)).
Highly abundant numbers: numbers k such that sigma(k) > sigma(m) for all m < k.
a(n) = LCM of denominators of Cotesian numbers {C(n,k), 0 <= k <= n}.
Primes congruent to 1 or 2 modulo 4; or, primes of form x^2 + y^2; or, -1 is a square mod p.
Numerators in Taylor series for tan(x). Also from Taylor series for tanh(x).
Numerators in Taylor series for cot x.
Coefficients of Legendre polynomials.
Coefficients of elliptic function sn.
Related to coefficient of m in Jacobi elliptic function cn(z, m).
Primes p == +- 3 (mod 8), or, primes p such that 2 is not a square mod p.
Binomial coefficient C(2n,n-4).
Binomial coefficient C(3n,n-1).
Untouchable numbers, also called nonaliquot numbers: impossible values for the sum of aliquot parts function (A001065).
Numerators of numbers occurring in continued fraction connected with expansion of gamma function.
Denominators of numbers occurring in continued fraction connected with expansion of gamma function.
Primes p such that (p+1)/2 is prime.
a(n) = (n-1)*n*(n+4)/6.
Sum of multinomial coefficients (n_1+n_2+...)!/(n_1!*n_2!*...) where (n_1, n_2, ...) runs over all integer partitions of n.
Expansion of Jacobi nome q in terms of parameter m/16.
Expansion of (theta_2(q)/theta_3(q))^4/16 in powers of q.
Coefficients of elliptic function cn.
Denominators of Bernoulli numbers B_0, B_1, B_2, B_4, B_6, ...
Denominator of (2n+1) B_{2n}, where B_n are the Bernoulli numbers.
Denominator of (2n+1)(2n+2) B_{2n}, where B_n are the Bernoulli numbers. Also denominators of the asymptotic expansion of the polygamma function psi”’(z).
Primes of form 8n+1, that is, primes congruent to 1 mod 8.
Primes == 3 (mod 8).
Primes of the form 8k + 5.
Final digit of prime(n).
From Euler’s Pentagonal Theorem: coefficient of q^n in Product_{m>=1} (1 - q^m).
Incomplete Gamma Function at -3.
a(n) = 2*n*a(n-1) + 1 with a(0) = 1.
Decimal expansion of zeta(5).
Decimal expansion of zeta(16).
Decimal expansion of zeta(18).
a(n) = floor( Gamma(n+1/2) ).
a(n) = floor( Gamma(n+1/3) ).
a(n) = floor( Gamma(n+2/3) ).
a(n) = floor( Gamma(n+1/4) ).
a(n) = floor( Gamma(n+3/4) ).
Nearest integer to Gamma(n + 1/3).
Nearest integer to Gamma(n+2/3).
Nearest integer to Gamma(n+1/4).
Nearest integer to Gamma(n+1/2).
Decimal expansion of log(4).
Decimal expansion of log(11).
Decimal expansion of log(12).
Decimal expansion of log(13).
Decimal expansion of log(17).
Decimal expansion of log(20).
Decimal expansion of log(21).
Decimal expansion of log(26).
Decimal expansion of log(78).
Decimal expansion of log(79).
Decimal expansion of log(80).
Decimal expansion of log(81).
Decimal expansion of log(82).
Decimal expansion of log(83).
Decimal expansion of log(84).
Decimal expansion of log(86).
Decimal expansion of log(87).
Decimal expansion of log(88).
Decimal expansion of log(89).
Decimal expansion of log(90).
Decimal expansion of log(91).
Decimal expansion of log(92).
Decimal expansion of log(93).
Decimal expansion of log(94).
Decimal expansion of log(95).
Decimal expansion of log(96).
Decimal expansion of log(97).
Decimal expansion of (-1)*Gamma’(1/2)/Gamma(1/2) where Gamma(x) denotes the Gamma function.
Juxtaposed partitions of 1,2,3,... into distinct parts, ordered by number of terms and then lexicographically.
Coefficients of Chebyshev polynomials of the first kind: triangle of coefficients in expansion of cos(n*x) in descending powers of cos(x).
Differences between consecutive odd primes, divided by 2.
Convert n from yards to meters.
Convert n from meters to yards.
Convert n from feet to meters.
Convert n from meters to feet.
Convert n from inches (”) to centimeters (cm).
Convert n from centimeters (cm) to inches (”).
Convert n from miles to kilometers (km).
Convert n from kilometers (km) to miles.
Convert n from pounds (lbs) to kilograms (kg).
Convert n from kilograms (kg) to pounds (lbs).
Convert n from degrees Celsius to Fahrenheit.
Convert n from degrees Fahrenheit to Centigrade (or Celsius).
Convert n from nautical miles to statute miles.
Convert n from statute miles to nautical miles.
Number of multisets associated with least integer of each prime signature.
Least integer of each prime signature, in graded (reflected or not) colexicographic order of exponents.
Triangle read by rows in which row n lists all the parts of all reversed partitions of n, sorted first by length and then lexicographically.
k appears partition(k) times.
Irregular triangle read by rows: row n (n >= 0) gives number of parts in all partitions of n (in Abramowitz and Stegun order).
Product of the lengths of the cycle types of the permutation created by duality and reversal on the partitions of n.
Denominators in Taylor series for cot x.
Denominators in Taylor series for tan(x).
Denominators in Taylor series for x * cosec(x).
Write cosec x = 1/x + Sum_{n>=1} e_n * x^(2n-1)/(2n-1)!; sequence gives numerators of e_n.
Write cosec x = 1/x + Sum e_n x^(2n-1)/(2n-1)!; sequence gives denominators of e_n.
Numerators of coefficients in Stirling’s expansion for log(Gamma(z)).
Denominators of coefficients in Stirling’s expansion for log(Gamma(z)).
Dirichlet inverse of the Jordan function J_2 (A007434).
Number of functions from a set to itself such that the sizes of the preimages of the individual elements in the range form the n-th partition in Abramowitz and Stegun order.
Triangle of coefficients of Chebyshev’s S(n,x) := U(n,x/2) polynomials (exponents in increasing order).
Numerators of numbers appearing in the Euler-Maclaurin summation formula.
Denominators of nonzero numbers appearing in the Euler-Maclaurin summation formula. (See A060054 for the definition of these numbers.)
Real half-period for the Weierstrass elliptic function with invariants g2=0, g3=1.
Triangle read by rows, in which row n lists the compositions of n in reverse lexicographic order.
Coefficient triangle of generalized Laguerre polynomials (a=1).
Triangle formed as follows: For n-th row, n >= 0, record the A000041(n) partitions of n; for each partition, write down number of ways to arrange the parts.
Indices of primes with primitive root 2.
Decimal expansion of Pi^2/12.
Denominators from e.g.f. 1/(1-exp(-x)) - 1/x.
Numerator of (2n+1)(2n+2) B_{2n}, where B_n are the Bernoulli numbers. Also numerators of the asymptotic expansion of the polygamma function psi”’(z).
Number of decimal digits of A070177(n).
Triangle of multinomial coefficients, read by rows (version 2).
Triangle in which n-th row lists all partitions of n, in graded reflected lexicographic order.
Triangle of coefficients of Chebyshev polynomials T_{2n+1} (x).
Array of number of partitions of n into m parts which have the parts of the partitions of m as exponents.
Characteristic array marking partitions of m whose parts are exponents of partitions of n into m parts.
Triangle read by rows. First in a series of triangular arrays counting permutations of partitions.
Decimal expansion of ”lemniscate case”.
Numbers n such that d(n) >= n-th harmonic number H(n)=sum_{i=1..n}1/i.
Numbers n such that, for some numbers (j,k), j<=k, n is the smallest positive multiple of j (or more) of the first k positive integers.
Numbers k such that, for all m < k, d_i(k) <= d_i(m) for i=1 to Min(d(k),d(m)), where d_i(k) denotes the i-th smallest divisor of k.
Let n be a number partitioned as n = b_1 + 2*b_2 + ... + n*b_n; then T(n) = (b_1)! * (b_2)! * ... (b_n)!. Irregular triangle read by rows, T(n, k) for n >= 1 and 1 <= k <= A000041(n).
Table read by rows: row n has a term T(n,k) for each of the partition(n) partitions of n. T(n,k) = binomial(n,m) where m is the number of parts.
Numerator of partial sums of a certain series. First member (m = 2) of a family.
Numerator of partial sums of a certain series.
Numerator of partial sums of a certain series.
Numerator of partial sums of a certain series.
Array of multinomial numbers (row reversed order of table A036039).
Numerator of Sum_{k=1..n} 1/k^6 = Zeta(6,n).
Irregular triangle T(n,m) (n >= 0) read by rows: row n lists numbers of distinct parts of partitions of n in Abramowitz-Stegun order.
Triangle, read by rows, where T(n,k) equals number of distinct partitions of triangular number n*(n+1)/2 into k different summands for n>=k>=1.
Number of distinct partitions of triangular numbers n*(n+1)/2.
Irregular triangle read by rows: T(n,k) is the Dyson’s rank of the k-th partition of n in Abramowitz-Stegun order.
a(n) = 2*n*a(n-1) - a(n-2), with a(0)=0, a(1)=1.
Semiprimes p*q where both p and q are primes of the form 6n-1 (A007528).
The matrix inverse of the unsigned Lah numbers A271703.
T(n,k) are coefficients used for power series inversion (sometimes called reversion), n >= 0, k = 1..A000041(n), read by rows.
Array used to obtain the complete symmetric function in n variables in terms of the elementary symmetric functions; irregular triangle T(n,k), read by rows, with n >= 1 and 1 <= k <= A000041(n).
Number of iterations of signature function required to get to [1] from partitions in Abramowitz and Stegun order.
Parity of partitions of n, with 1 for even, 0 for odd (!). The definition follows.
Signature of partitions in Abramowitz and Stegun order.
Table of Durfee square of partitions in Abramowitz and Stegun order.
Maximum rectangle of partitions in Abramowitz and Stegun order.
Number of subpartitions of partitions in Abramowitz and Stegun order.
Expansion of elliptic modular function lambda in powers of the nome q.
Array of product of parts of the partitions of n with only prime parts.
Characteristic array for partitions with only prime parts.
Irregular triangle read by rows: dimensions of the irreducible representations of the symmetric group S_n.
Lengths of partitions into distinct parts in Abramowitz and Stegun order.
Product of parts in n-th partition in Abramowitz and Stegun order.
Distribution of A063834 in Abramowitz and Stegun order.
Distribution of A060642 in Abramowitz and Stegun order.
Expansion of q^2 in powers of m/16 where q is Jacobi nome and m is the parameter.
Numerators of expansion of original Debye function D(3,x).
Numerators of expansion for Debye function for n=2: D(2,x).
Numerators of expansion of Debye function for n=4: D(4,x).
Triangle read by rows: T(n,k) = binomial(2*n+1,k), 0 <= k <= n.
Triangle of coefficients of (1 - x)^n*B_n(x/(1 - x)), where B_n(x) is the n-th Bell polynomial.
Number of distributive sublattices of the lattice of k-tuples less than the n-th partition (in Abramowitz and Stegun order), that include the maximum element.
Number of distributive sublattices of the lattice of k-tuples less than the n-th partition (in Abramowitz and Stegun order).
Triangle of coefficients of n!*(1 - x)^n*L_n(x/(1 - x)), where L_n(x) is the Laguerre polynomial.
Triangle read by rows: coefficients of expansion in powers of x of the polynomials defined by p(n, x) = (2*n - 1)*p(n - 1, x) + (n - 1)^2*x^2*p(n - 2, x).
Table with all compositions sorted first by total, then by length and finally lexicographically.
One half of even powers of 2*x in terms of Chebyshev’s T-polynomials.
Coefficient table for Chebyshev polynomials T(2*n,x) (increasing even powers x, without zeros).
Numerators of partial sums of a series for the inverse of the arithmetic-geometric mean (agM) of sqrt(2)/2 and 1.
Numerators of partial sums of a series for the inverse of the arithmetic-geometric mean (AGM) of sqrt(3)/2 and 1.
Numerators of partial sums of a series for the inverse of the arithmetic-geometric mean (agM) of 1/2 and 1.
Numerators of partial sums of a series for the inverse of the arithmetic-geometric mean (agM) of 2/sqrt(5) and 1.
Numbers associated to partitions, used for combinatoric interpretation of Lah triangle numbers A105278; elementary Schur polynomials / functions.
Triangular table of denominators of the coefficients of Laguerre-Sonin polynomials L(1/2,n,x).
Triangular table of coefficients of Laguerre-Sonin polynomials n!*2^n*Lag(n,x/2,1/2) of order 1/2.
Coefficients of first difference of Chebyshev S polynomials.
Decimal expansion of the square of Pi*log_10(e).
Triangular table of numerators of the coefficients of Laguerre-Sonin polynomials L(1/2,n,x).
A certain partition array in Abramowitz-Stegun order (A-St order).
A certain partition array in Abramowitz-Stegun order (A-St order).
A certain partition array in Abramowitz-Stegun order (A-St order), called M_3(3)/M_3.
A certain partition array in Abramowitz-Stegun (A-St) order.
A certain partition array in Abramowitz-Stegun order (A-St order), called M_3(4)/M_3.
Coefficients T(j, k) of a partition transform for Lagrange compositional inversion of a function or generating series in terms of the coefficients of the power series for its reciprocal. Enumeration of noncrossing partitions and primitive parking functions. T(n,k) for n >= 1 and 1 <= k <= A000041(n-1), an irregular triangle read by rows.
A certain partition array in Abramowitz-Stegun order (A-St order), called M_3(5).
A certain partition array in Abramowitz-Stegun order (A-St order), called M_3(5)/M_3.
A certain partition array in Abramowitz-Stegun order (A-St order), called M_3(6).
A certain partition array in Abramowitz-Stegun order (A-St order), called M_3(6)/M_3.
A certain partition array in Abramowitz-Stegun (A-St)order, called M_0(3).
A certain partition array in Abramowitz-Stegun order (A-St order), called M_0(3)/M_0.
Characteristic sequence for sequence A026905.
Irregular triangle read by rows: coefficients C(j,k) of a partition transform for direct Lagrange inversion.
Integration of A053120: triangle of coefficients of integration of Chebyshev’s T(n,x) polynomials (powers of x in increasing order).
Triangle read by rows: numerators of coefficients of the Debye-type polynomial u_n used for asymptotic Airy-type expansions of Bessel functions of arbitrary large order.
Triangle read by rows: numerators of coefficients of the Debye-type polynomial v_n used for asymptotic Airy-type expansions of Bessel functions of arbitrarily large order.
Integers n such that sigma_2(n) = sigma_2(n + 2) where sigma_2(n) is the sum of squares of divisors of n (A001157).
Numbers n such that phi(n)/n = 16/29.
Numbers n such that phi(phi(n)) + sigma(sigma(n)) is an 8th power.
Numbers k such that tau(phi(k))= sopf(k).
Numbers k such that tau(phi(k)) = phi(sum-of-prime-divisors(k)).
Numbers k such that tau(phi(k)) = sigma(sopf(k)).
Numbers n such that rad(n)^2 divides sigma(n).
Numbers n such that phi(tau(n))= rad(n)
Numbers k such that tau(phi(k)) = rad(k).
Numbers k such that phi(tau(k)) = tau(rad(k)).
Numbers n such that tau(phi(n))= phi(rad(n))
Numbers k such that phi(phi(k)) = sigma(rad(k)).
a(n) is the period k such that binomial(m, n) (mod 10) = binomial(m + k, n) (mod 10).
a(n) = n - phi(2*n), where phi() is the Euler totient A000010().
a(n) = n*(n-3)*(n^2-7*n+14)/8.
Characteristic array for partitions which define multiset repetition classes.
Inversion of e.g.f. formal power series. Partition array in Abramowitz-Stegun (A-St) order.
a(n) is the number of iterations of f(n) = n-phi(tau(n)) needed to reach 1.
Numerators of exponential transform of 1/n.
a(n) = binomial(n^2,n+1)/n.
Decimal expansion of Sum_{k>=1} 1/2^Partition(k).
Partitions in Abramowitz-Stegun order A036036 mapped one-to-one to positive integers.
Row n of table represents second signature of n: list of exponents >= 2 in canonical prime factorization of n, in nonincreasing order, or 0 if no such exponent exists.
Decimal expansion of largest x such that x^2 = Gamma(x+1).
Smallest prime of the form ChebyshevT[2^n, x].
Smallest number k such that LegendreP[2*n, k] is prime.
Smallest prime of the form LegendreP[2*n, k], k integer > 0.
Irregular triangular array read by rows: row n gives a list of the partitions of n into distinct Fibonacci numbers. The order of the partitions is like in Abramowitz-Stegun.
Numerators for partial sums of dilog(1/2).
Decimal expansion of dilog(phi-1) = polylog(2, 2-phi) with phi = (1 + sqrt(5))/2.
Decimal expansion of -dilog(phi) = -polylog(2, 1-phi) with phi = (1 + sqrt(5))/2.
a(n+6) = 6*a(n+4) - 12*a(n+2) + 8*a(n), a(0)..a(5) = 8,0,9,0,8,0.
Triangle read by rows: Lagrange (compositional) inversion of a function in terms of the coefficients of the Taylor series expansion of its reciprocal, scaled version of A248927, n >= 1, k = 1..A000041(n-1).
Triangle read by rows: T(n,k) are the coefficients of the Lagrange (compositional) inversion of a function in terms of the Taylor series expansion of its reciprocal, n >= 1, k = 1..A000041(n-1).
Coefficients of reduced partition polynomials of A134264 for computing Lagrange compositional inversion.
Triangle read by rows: coefficients of the partition polynomials for the reciprocal of the derivative of a power series, g(x)= 1/h’(x).
Triangle read by rows in which the n-th row lists the multinomials A036038 for all partitions of 2n with only even parts in Abramowitz-Stegun ordering.
Irregular triangle read by rows in which the n-th row lists multinomials (A036040) for partitions of 2n which have only even parts in Abramowitz-Stegun ordering.
Partition array for the products of the hook lengths of Ferrers (Young) diagrams corresponding to the partitions of n, written in Abramowitz-Stegun order.
Coefficients of the Faber partition polynomials.
Irregular triangle read by rows, giving the numerators of the coefficients of the Eisenstein series G_{2*n} multiplied by 2*n-1, for n >= 2. Also Laurent coefficients of Weierstrass’s P function.
Coefficients in the expansion of q^(1/2) in odd powers of k/4, where q is the Jacobi nome and k^2 the parameter of elliptic functions. Also coefficients in the expansion of q in odd powers of (1/4)*(1 - k’) / (1 + k’) with k’^2 the complementary parameter.
Irregular triangle read by rows: T(n, k) gives the coefficients of the Girard-Waring formula for the sum of n-th power of five indeterminates in terms of their elementary symmetric functions (reverse Abramowitz-Stegun order of partitions).
Partition array T(n, k) for the coefficients of the n-th power sums of the second elementary symmetric function in terms of the elementary symmetric functions.
Coefficients of successive polynomials formed by iterating f(x) = -1 + 2x^2. Irregular triangle read by rows.
Partition array giving in row n, for n >= 1, the coefficients of the Witt symmetric function w_n in terms of the elementary symmetric functions (using partitions in the Abramowitz-Stegun order).
Partition array giving in row n, for n >= 1, the coefficients of the Witt symmetric function w_n, multiplied by n!, in terms of the power sum symmetric functions (using partitions in the Abramowitz-Stegun order)
Signed version of the partition array A036039 (signed M_2 multinomial numbers).
Decimal expansion of (7/120)*Pi^4 = (21/4)*zeta(4).
Irregular triangle read by rows: Row p gives number of non-overlapping clusters of 2q-plets joining 2p points on a circle, i.e., number of noncrossing partitions from A134264 with h_k for k odd replaced by zero.
Irregular triangle read by rows: Refined 3-Narayana triangle. Coefficients of partition polynomials of A134264, a refined Narayana triangle enumerating noncrossing partitions, with all h_k other than h_0, h_3, h_6, ..., h_(3n), ... replaced by zero.