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Motzkin numbers

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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: 26.6 Other Lattice Path Numbers
§26.6 Other Lattice Path Numbers
Motzkin Number M ( n )
M ( n ) is the number of lattice paths from ( 0 , 0 ) to ( n , n ) that stay on or above the line y = x and are composed of directed line segments of the form ( 2 , 0 ) , ( 0 , 2 ) , or ( 1 , 1 ) . …
Table 26.6.2: Motzkin numbers M ( n ) .
n M ( n ) n M ( n ) n M ( n ) n M ( n ) n M ( n )
§26.6(iv) Identities
3: 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 . …
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: 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
6: 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
7: 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
8: 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
9: 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
10: 24.19 Methods of Computation
§24.19(i) Bernoulli and Euler Numbers and Polynomials
Equations (24.5.3) and (24.5.4) enable B n and E n to be computed by recurrence. …A similar method can be used for the Euler numbers based on (4.19.5). …
§24.19(ii) Values of B n Modulo p
We list here three methods, arranged in increasing order of efficiency. …