Catalan numbers ♦ 4 matching pages ♦ SearchAdvancedHelp (0.001 seconds) 4 matching pages 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 ) , ⓘ Symbols: C ( n ) : Catalan number, k : nonnegative integer and n : nonnegative integer Permalink: http://dlmf.nist.gov/26.5.E3 Encodings: pMML, png, TeX See also: Annotations for §26.5(iii), §26.5 and Ch.26 … 2: 26.6 Other Lattice Path Numbers … ► §26.6(iv) Identities ► 26.6.12 C ( n ) = ∑ k = 1 n N ( n , k ) , ⓘ Symbols: C ( n ) : Catalan number, k : nonnegative integer, n : nonnegative integer and N ( n , k ) : Narayana number Permalink: http://dlmf.nist.gov/26.6.E12 Encodings: pMML, png, TeX See also: Annotations for §26.6(iv), §26.6 and Ch.26 ► 26.6.13 M ( n ) = ∑ k = 0 n ( − 1 ) k ( n k ) C ( n + 1 − k ) , ⓘ Symbols: C ( n ) : Catalan number, ( m n ) : binomial coefficient, k : nonnegative integer, n : nonnegative integer and M ( n ) : Motzkin number Permalink: http://dlmf.nist.gov/26.6.E13 Encodings: pMML, png, TeX See also: Annotations for §26.6(iv), §26.6 and Ch.26 ► 26.6.14 C ( n ) = ∑ k = 0 2 n ( − 1 ) k ( 2 n k ) M ( 2 n − k ) . ⓘ Symbols: C ( n ) : Catalan number, ( m n ) : binomial coefficient, k : nonnegative integer, n : nonnegative integer and M ( n ) : Motzkin number Permalink: http://dlmf.nist.gov/26.6.E14 Encodings: pMML, png, TeX See also: Annotations for §26.6(iv), §26.6 and Ch.26 3: 26.1 Special Notation … ► ► ( m n ) binomial coefficient. … ► C ( n ) Catalan number. … ► … 4: 25.11 Hurwitz Zeta Function … ►where H n are the harmonic numbers: ► 25.11.33 H n = ∑ k = 1 n k − 1 . ⓘ Defines: H n : harmonic number Symbols: k : nonnegative integer, n : nonnegative integer and h ( n ) : harmonic number Keywords: harmonic number Source: Knuth (1968, Section 1.2.7, p. 73) Referenced by: §25.11(viii), §25.16(ii), Erratum (V1.1.4) for Notation Permalink: http://dlmf.nist.gov/25.11.E33 Encodings: pMML, png, TeX Notation (effective with 1.1.4): The notation previously used for the harmonic number h ( n ) has been replaced to be H n . Suggested 2021-08-23 by Gergő Nemes See also: Annotations for §25.11(viii), §25.11 and Ch.25 … ► 25.11.39 ∑ k = 2 ∞ k 2 k ζ ( k + 1 , 3 4 ) = 8 G , ⓘ Symbols: ζ ( s , a ) : Hurwitz zeta function, k : nonnegative integer and G : Catalan’s constant Keywords: Catalan’s constant, infinite series Source: Adamchik and Srivastava (1998, (2.30), p. 138) Permalink: http://dlmf.nist.gov/25.11.E39 Encodings: pMML, png, TeX See also: Annotations for §25.11(xi), §25.11 and Ch.25 ►where G is Catalan’s constant: ► 25.11.40 G ≡ ∑ n = 0 ∞ ( − 1 ) n ( 2 n + 1 ) 2 = 0.91596 55941 772 … . ⓘ Symbols: ≡ : equals by definition, n : nonnegative integer and G : Catalan’s constant Keywords: Catalan’s constant, definition Source: Adamchik and Srivastava (1998, (2.32), p. 139) Notes: For more digits see OEIS Sequence A006752; see also Sloane (2003). Permalink: http://dlmf.nist.gov/25.11.E40 Encodings: pMML, png, TeX See also: Annotations for §25.11(xi), §25.11 and Ch.25 …