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Bernoulli Numbers and Polynomials►The origin of the notation , , 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 , , as defined in §24.2(ii), were used in Lucas (1891) and Nörlund (1924). …
§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…
Multiplicative Number Theory.
3rd edition, Graduate Texts in Mathematics, Vol. 74, Springer-Verlag, New York.
Elements of the Theory of Numbers.
Harcourt/Academic Press, San Diego, CA.
Sums of products of Bernoulli numbers.
J. Number Theory 60 (1), pp. 23–41.
Bernoulli Numbers and Confluent Hypergeometric Functions.
In Number Theory for the Millennium, I (Urbana, IL, 2000),
A combinatorial interpretation of the Seidel generation of Genocchi numbers.
Ann. Discrete Math. 6, pp. 77–87.
… ► denotes the number of compositions of , and is the number of compositions into exactly parts. is the number of compositions of with no 1’s, where again . … ►
26.11.1… ►The Fibonacci numbers are determined recursively by … ►Additional information on Fibonacci numbers can be found in Rosen et al. (2000, pp. 140–145).
§27.18 Methods of Computation: Primes►An overview of methods for precise counting of the number of primes not exceeding an arbitrary integer 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, , and Fermat numbers, . …
§26.6 Other Lattice Path Numbers… ►
Delannoy Number… ►
Motzkin Number… ►
Narayana Number… ►
§26.5 Lattice Paths: Catalan Numbers►
§26.5(i) Definitions► is the Catalan number. … ►
§26.5(ii) Generating Function… ►
§26.5(iii) Recurrence Relations…
… ►As an example, is an element of The inversion number is the number of pairs of elements for which the larger element precedes the smaller: … ► ►The Eulerian number, denoted , is the number of permutations in with exactly descents. …The Eulerian number is equal to the number of permutations in with exactly excedances. … ►