Abate, Joseph; Whitt, Ward Brownian motion and the generalized Catalan numbers. (English) Zbl 1229.11028 J. Integer Seq. 14, No. 2, Article 11.2.6, 15 p. (2011). The authors present formulas with generalized Catalan numbers. Reviewer: Florin Nicolae (Berlin) Cited in 2 Documents MSC: 11B65 Binomial coefficients; factorials; \(q\)-identities Keywords:generalized Catalan numbers Software:OEIS PDFBibTeX XMLCite \textit{J. Abate} and \textit{W. Whitt}, J. Integer Seq. 14, No. 2, Article 11.2.6, 15 p. (2011; Zbl 1229.11028) Full Text: EuDML EMIS Online Encyclopedia of Integer Sequences: Central binomial coefficients: binomial(2*n,n) = (2*n)!/(n!)^2. a(n) = binomial(2*n+1, n+1): number of ways to put n+1 indistinguishable balls into n+1 distinguishable boxes = number of (n+1)-st degree monomials in n+1 variables = number of monotone maps from 1..n+1 to 1..n+1. Triangle read by rows. T(n, k) = binomial(n+k-1, k) for 0 <= k <= n. Generalized Catalan numbers C(2; n). Generalized Catalan numbers C(3; n). Generalized Catalan numbers C(2,2; n). Generalized Catalan numbers C(3,3; n). Generalized Catalan numbers C(4,4; n). Generalized Catalan numbers C(5,5; n). Generalized Catalan numbers C(6,6; n). Generalized Catalan numbers C(7,7; n). Generalized Catalan numbers C(8,8; n). Generalized Catalan numbers C(9,9; n). Expansion of 1/(1-2*x*c(3*x)), c(x) the g.f. of A000108. Convolution of generalized Catalan numbers A064062 (called C(n;2)). Convolution of generalized Catalan sequence A064063 (named C(3;n)). Generalized Catalan numbers C(2,3;n)=C(3,2;n). Generalized Catalan numbers C(2,4;n)=C(4,2;n). Central terms of the triangle in A119258. Dot product of the rows of triangle A046899 with vector (1,2,4,8,...) (= A000079). A(n, k) = hypergeometric([-k, k+1], [-k-1], n), square array read by ascending antidiagonals for n,k >= 0. a(2n+1) = a(n) for n >= 0, a(2n) = a(n) + a(n - 2^A007814(n)) for n > 0 with a(0) = 1.