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%I #26 Apr 01 2020 22:18:31
%S 1,8,47,244,1186,5536,25147,112028,491870,2135440,9188406,39249768,
%T 166656772,704069248,2961699667,12412521388,51854046982,216013684528,
%U 897632738722,3721813363288,15401045060572,63616796642368,262357557683422,1080387930269464
%N A sum with next-to-central binomial coefficients of even order, Catalan related.
%C Proof by induction.
%C 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
%C . _|.....|
%C _|.....__|
%C The first has area 1 below it, the second area 0 and so a(0)=1. - _David Callan_, Dec 09 2004
%C Convolution of A000346 with A001700. - _Philippe Deléham_, May 19 2009
%H Michael De Vlieger, <a href="/A029760/b029760.txt">Table of n, a(n) for n = 0..1657</a>
%H Ran Pan, Jeffrey B. Remmel, <a href="http://arxiv.org/abs/1601.07988">Paired patterns in lattice paths</a>, arXiv:1601.07988 [math.CO], 2016.
%H Sittipong Thamrongpairoj, <a href="https://escholarship.org/uc/item/7j561211">Dowling Set Partitions, and Positional Marked Patterns</a>, Ph. D. Dissertation, University of California-San Diego (2019).
%F 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).
%F 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).
%F a(n) = Sum_{k=0..n+1} A039598(n+1,k)*k^2. - _Philippe Deléham_, Dec 16 2007
%t a[n_] := (n+3)^2 CatalanNumber[n+2]/2 - 2^(2n+3);
%t Table[a[n], {n, 0, 23}] (* _Jean-François Alcover_, Sep 25 2018 *)
%Y Cf. A000108, A002457, A008549, A000984, A139262.
%K nonn
%O 0,2
%A _Wolfdieter Lang_
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