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 A029760 A sum with next-to-central binomial coefficients of even order, Catalan related. 10
 1, 8, 47, 244, 1186, 5536, 25147, 112028, 491870, 2135440, 9188406, 39249768, 166656772, 704069248, 2961699667, 12412521388, 51854046982, 216013684528, 897632738722, 3721813363288, 15401045060572, 63616796642368, 262357557683422, 1080387930269464 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS Proof by induction. a(n) = total area below paths consisting of steps east (1,0) and north (0,1) from (0,0) to (n+2,n+2) that stay weakly below y=x. For example, the two paths with n=0 are . _|.....| _|.....__| The first has area 1 below it, the second area 0 and so a(0)=1. - David Callan, Dec 09 2004 Convolution of A000346 with A001700. - Philippe Deléham, May 19 2009 LINKS Michael De Vlieger, Table of n, a(n) for n = 0..1657 Ran Pan, Jeffrey B. Remmel, Paired patterns in lattice paths, arXiv:1601.07988 [math.CO], 2016. Sittipong Thamrongpairoj, Dowling Set Partitions, and Positional Marked Patterns, Ph. D. Dissertation, University of California-San Diego (2019). FORMULA a(n) = 4^(n+1)*Sum_{k=1..n+1} binomial(2k, k-1)/4^k = ((n+3)^2)*C(n+2)/2-2^(2*n+3), C = Catalan. Also a(n+1)=4*a(n)+binomial(2(n+2), n+1). G.f.: (d/dx)c(x)/(1-4*x), where c(x) = g.f. for Catalan numbers; convolution of A001791 and powers of 4. G.f. also c(x)^2/(1-4*x)^(3/2); convolution of Catalan numbers A000108 C(n), n >= 1, with A002457; convolution of A008549(n), n >= 1, with A000984 (central binomial coefficients). a(n) = Sum_{k=0..n+1} A039598(n+1,k)*k^2. - Philippe Deléham, Dec 16 2007 MATHEMATICA a[n_] := (n+3)^2 CatalanNumber[n+2]/2 - 2^(2n+3); Table[a[n], {n, 0, 23}] (* Jean-François Alcover, Sep 25 2018 *) CROSSREFS Cf. A000108, A002457, A008549, A000984, A139262. Sequence in context: A099110 A106393 A300167 * A139262 A026900 A016198 Adjacent sequences: A029757 A029758 A029759 * A029761 A029762 A029763 KEYWORD nonn AUTHOR STATUS approved

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Last modified April 1 12:16 EDT 2023. Contains 361691 sequences. (Running on oeis4.)