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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
  • 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.
  • B. Döring (1966) Complex zeros of cylinder functions. Math. Comp. 20 (94), pp. 215–222.
  • D. Dumont and G. Viennot (1980) A combinatorial interpretation of the Seidel generation of Genocchi numbers. Ann. Discrete Math. 6, pp. 77–87.
  • T. M. Dunster (1989) Uniform asymptotic expansions for Whittaker’s confluent hypergeometric functions. SIAM J. Math. Anal. 20 (3), pp. 744–760.
  • 4: 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
    5: 27.15 Chinese Remainder Theorem
    §27.15 Chinese Remainder Theorem
    This theorem is employed to increase efficiency in calculating with large numbers by making use of smaller numbers in most of the calculation. …Their product m has 20 digits, twice the number of digits in the data. …These numbers, in turn, are combined by the Chinese remainder theorem to obtain the final result ( mod m ) , which is correct to 20 digits. …
    6: 27.2 Functions
    where p 1 , p 2 , , p ν ( n ) are the distinct prime factors of n , each exponent a r is positive, and ν ( n ) is the number of distinct primes dividing n . …Euclid’s Elements (Euclid (1908, Book IX, Proposition 20)) gives an elegant proof that there are infinitely many primes. … (See Gauss (1863, Band II, pp. 437–477) and Legendre (1808, p. 394).) …
    §27.2(ii) Tables
    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: 24.20 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. …
    9: 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
    10: 26.13 Permutations: Cycle Notation
    The Stirling cycle numbers of the first kind, denoted by [ n k ] , count the number of permutations of { 1 , 2 , , n } with exactly k cycles. They are related to Stirling numbers of the first kind by …See §26.8 for generating functions, recurrence relations, identities, and asymptotic approximations. … The derangement number, d ( n ) , is the number of elements of 𝔖 n with no fixed points: … A permutation is even or odd according to the parity of the number of transpositions. …