Delannoy numbers
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11: 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. …12: 24.10 Arithmetic Properties
§24.10 Arithmetic Properties
… ►Here and elsewhere two rational numbers are congruent if the modulus divides the numerator of their difference. ►§24.10(ii) Kummer Congruences
… ►§24.10(iii) Voronoi’s Congruence
… ►§24.10(iv) Factors
…13: 26.21 Tables
§26.21 Tables
►Abramowitz and Stegun (1964, Chapter 24) tabulates binomial coefficients for up to 50 and up to 25; extends Table 26.4.1 to ; tabulates Stirling numbers of the first and second kinds, and , for up to 25 and up to ; tabulates partitions and partitions into distinct parts for up to 500. ►Andrews (1976) contains tables of the number of unrestricted partitions, partitions into odd parts, partitions into parts , partitions into parts , and unrestricted plane partitions up to 100. It also contains a table of Gaussian polynomials up to . ►Goldberg et al. (1976) contains tables of binomial coefficients to and Stirling numbers to .14: 24.14 Sums
§24.14 Sums
►§24.14(i) Quadratic Recurrence Relations
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24.14.2
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§24.14(ii) Higher-Order Recurrence Relations
… ►For other sums involving Bernoulli and Euler numbers and polynomials see Hansen (1975, pp. 331–347) and Prudnikov et al. (1990, pp. 383–386).15: 26.1 Special Notation
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►Other notations for , the Stirling numbers of the first kind, include (Abramowitz and Stegun (1964, Chapter 24), Fort (1948)), (Jordan (1939), Moser and Wyman (1958a)), (Milne-Thomson (1933)), (Carlitz (1960), Gould (1960)), (Knuth (1992), Graham et al. (1994), Rosen et al. (2000)).
►Other notations for , the Stirling numbers of the second kind, include (Fort (1948)), (Jordan (1939)), (Moser and Wyman (1958b)), (Milne-Thomson (1933)), (Carlitz (1960), Gould (1960)), (Knuth (1992), Graham et al. (1994), Rosen et al. (2000)), and also an unconventional symbol in Abramowitz and Stegun (1964, Chapter 24).
binomial coefficient. | |
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Eulerian number. | |
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Bell number. | |
Catalan number. | |
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16: 27.3 Multiplicative Properties
§27.3 Multiplicative Properties
►Except for , , , and , the functions in §27.2 are multiplicative, which means and … ►
27.3.2
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27.3.6
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27.3.10
17: 27.1 Special Notation
§27.1 Special Notation
… ►positive integers (unless otherwise indicated). | |
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prime numbers (or primes): integers () with only two positive integer divisors, and the number itself. | |
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real numbers. | |
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18: 27.12 Asymptotic Formulas: Primes
§27.12 Asymptotic Formulas: Primes
… ►Prime Number Theorem
… ►The number of such primes not exceeding is … ►There are infinitely many Carmichael numbers.19: 27.2 Functions
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