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Catalan numbers

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1: 26.5 Lattice Paths: Catalan Numbers
C ( n ) is the Catalan number. …
Table 26.5.1: Catalan numbers.
n C ( n ) n C ( n ) n C ( n )
26.5.3 C ( n + 1 ) = k = 0 n C ( k ) C ( n - k ) ,
2: 26.6 Other Lattice Path Numbers
§26.6(iv) Identities
26.6.12 C ( n ) = k = 1 n N ( n , k ) ,
26.6.13 M ( n ) = k = 0 n ( - 1 ) k ( n k ) C ( n + 1 - k ) ,
26.6.14 C ( n ) = k = 0 2 n ( - 1 ) k ( 2 n k ) M ( 2 n - k ) .
3: 26.1 Special Notation
( m n )

binomial coefficient.

C ( n )

Catalan number.

4: 25.11 Hurwitz Zeta Function
25.11.22 ζ ( 1 - 2 n , 1 2 ) = - B 2 n ln 2 n 4 n - ( 2 2 n - 1 - 1 ) ζ ( 1 - 2 n ) 2 2 n - 1 , n = 1 , 2 , 3 , .
25.11.33 h ( n ) = k = 1 n k - 1 .
25.11.39 k = 2 k 2 k ζ ( k + 1 , 3 4 ) = 8 G ,
where G is Catalan’s constant:
25.11.40 G n = 0 ( - 1 ) n ( 2 n + 1 ) 2 = 0.91596 55941 772 .