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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: Bibliography C
  • L. Carlitz (1953) Some congruences for the Bernoulli numbers. Amer. J. Math. 75 (1), pp. 163–172.
  • L. Carlitz (1954a) q -Bernoulli and Eulerian numbers. Trans. Amer. Math. Soc. 76 (2), pp. 332–350.
  • L. Carlitz (1954b) A note on Euler numbers and polynomials. Nagoya Math. J. 7, pp. 35–43.
  • L. Carlitz (1958) Expansions of q -Bernoulli numbers. Duke Math. J. 25 (2), pp. 355–364.
  • J. R. Cash and R. V. M. Zahar (1994) A Unified Approach to Recurrence Algorithms. In Approximation and Computation (West Lafayette, IN, 1993), R. V. M. Zahar (Ed.), International Series of Computational Mathematics, Vol. 119, pp. 97–120.
  • 3: DLMF Project News
    error generating summary
    4: 3.6 Linear Difference Equations
    For further information see Wimp (1984, Chapters 7–8), Cash and Zahar (1994), and Lozier (1980).
    5: 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).
    6: 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 . …
    7: 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
    8: 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
    9: 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
    10: 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