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A029760
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A sum with next-to-central binomial coefficients of even order, Catalan related.
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10
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1, 8, 47, 244, 1186, 5536, 25147, 112028, 491870, 2135440, 9188406, 39249768, 166656772, 704069248, 2961699667, 12412521388, 51854046982, 216013684528, 897632738722, 3721813363288, 15401045060572, 63616796642368, 262357557683422, 1080387930269464
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OFFSET
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0,2
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COMMENTS
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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
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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
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LINKS
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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).
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FORMULA
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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
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MATHEMATICA
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a[n_] := (n+3)^2 CatalanNumber[n+2]/2 - 2^(2n+3);
Table[a[n], {n, 0, 23}] (* Jean-François Alcover, Sep 25 2018 *)
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CROSSREFS
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Cf. A000108, A002457, A008549, A000984, A139262.
Sequence in context: A099110 A106393 A300167 * A139262 A026900 A016198
Adjacent sequences: A029757 A029758 A029759 * A029761 A029762 A029763
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KEYWORD
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nonn
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AUTHOR
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Wolfdieter Lang
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STATUS
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approved
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