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1: 24.1 Special Notation
Bernoulli Numbers and Polynomials
The origin of the notation B n , B n ( x ) , 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 ( x ) , as defined in §24.2(ii), were used in Lucas (1891) and Nörlund (1924). …
2: 24.15 Related Sequences of Numbers
§24.15 Related Sequences of Numbers
§24.15(i) Genocchi Numbers
§24.15(ii) Tangent Numbers
§24.15(iii) Stirling Numbers
§24.15(iv) Fibonacci and Lucas Numbers
3: Bibliography D
  • H. Davenport (2000) Multiplicative Number Theory. 3rd edition, Graduate Texts in Mathematics, Vol. 74, Springer-Verlag, New York.
  • J. B. Dence and T. P. Dence (1999) Elements of the Theory of Numbers. Harcourt/Academic Press, San Diego, CA.
  • K. Dilcher (1996) Sums of products of Bernoulli numbers. J. Number Theory 60 (1), pp. 23–41.
  • K. Dilcher (2002) Bernoulli Numbers and Confluent Hypergeometric Functions. In Number Theory for the Millennium, I (Urbana, IL, 2000), pp. 343–363.
  • D. Dumont and G. Viennot (1980) A combinatorial interpretation of the Seidel generation of Genocchi numbers. Ann. Discrete Math. 6, pp. 77–87.
  • 4: 26.11 Integer Partitions: Compositions
    c ( n ) denotes the number of compositions of n , and c m ( n ) is the number of compositions into exactly m parts. c ( T , n ) is the number of compositions of n with no 1’s, where again T = { 2 , 3 , 4 , } . …
    26.11.1 c ( 0 ) = c ( T , 0 ) = 1 .
    The Fibonacci numbers are determined recursively by … Additional information on Fibonacci numbers can be found in Rosen et al. (2000, pp. 140–145).
    5: 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 . …
    6: 26.6 Other Lattice Path Numbers
    §26.6 Other Lattice Path Numbers
    Delannoy Number D ( m , n )
    Motzkin Number M ( n )
    Narayana Number N ( n , k )
    §26.6(iv) Identities
    7: 26.5 Lattice Paths: Catalan Numbers
    §26.5 Lattice Paths: Catalan Numbers
    §26.5(i) Definitions
    C ( n ) is the Catalan number. …
    §26.5(ii) Generating Function
    §26.5(iii) Recurrence Relations
    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 n k , is the number of permutations in 𝔖 n with exactly k descents. …The Eulerian number n k is equal to the number of permutations in 𝔖 n with exactly k excedances. …
    §26.14(iii) Identities
    9: 26.7 Set Partitions: Bell Numbers
    §26.7 Set Partitions: Bell Numbers
    §26.7(i) Definitions
    §26.7(ii) Generating Function
    §26.7(iii) Recurrence Relation
    §26.7(iv) Asymptotic Approximation
    10: 26.8 Set Partitions: Stirling Numbers
    §26.8 Set Partitions: Stirling Numbers
    §26.8(i) Definitions
    §26.8(v) Identities
    §26.8(vi) Relations to Bernoulli Numbers