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%I #37 Apr 11 2024 10:33:58
%S 1,5,20,76,286,1078,4081,15521,59279,227239,873885,3370029,13027729,
%T 50469889,195892564,761615284,2965576714,11563073314,45141073924,
%U 176423482324,690215089744,2702831489824,10593202603774,41550902139550,163099562175850,640650742051802
%N a(n) = Sum_{k=0..n} C(2k+2,k).
%H Vincenzo Librandi, <a href="/A057552/b057552.txt">Table of n, a(n) for n = 0..1000</a>
%H Jean-Luc Baril, Pamela E. Harris, Kimberly J. Harry, Matt McClinton, and José L. Ramírez, <a href="https://arxiv.org/abs/2404.05672">Enumerating runs, valleys, and peaks in Catalan words</a>, arXiv:2404.05672 [math.CO], 2024. See p. 21.
%H Jean-Luc Baril, Sergey Kirgizov, and Mehdi Naima, <a href="https://arxiv.org/abs/2309.00426">A lattice on Dyck paths close to the Tamari lattice</a>, arXiv:2309.00426 [math.CO], 2023.
%H A. V. Kitaev and A. Vartanian, <a href="https://arxiv.org/abs/2304.05671">Algebroid Solutions of the Degenerate Third Painlevé Equation for Vanishing Formal Monodromy Parameter</a>, arXiv:2304.05671 [math.CA], 2023. See p. 59.
%F G.f.: 1/2*(2*x+(1-4*x)^(1/2)-1)/(1-4*x)^(1/2)/x^2/(-1+x). - _Vladeta Jovovic_, Sep 10 2003
%F D-finite with recurrence: n*(n+2)*a(n) = (5*n^2+8*n+2)*a(n-1) - 2*(n+1)*(2*n+1)*a(n-2). - _Vaclav Kotesovec_, Oct 11 2012
%F a(n) ~ 2^(2*n+4)/(3*sqrt(Pi*n)). - _Vaclav Kotesovec_, Oct 11 2012
%F a(n) = Sum_{k=1..n+1} k*A000108(k) = Sum_{k=1..n+1} A001791(k) = (A000108(n+1) * (4*n + 6 - (n+2)*hypergeom([1,-n-1], [-n-1/2], 1/4]) - 1)/2.
%F a(n) = Sum_{k=1..n+1} Sum_{i=1..k} C(i+k-1,k). - _Wesley Ivan Hurt_, Sep 19 2017
%p a:= n->add(binomial(2*j+2, j), j=0..n): seq(a(n), n=0..24); # _Zerinvary Lajos_, Oct 25 2006
%t Table[Sum[Binomial[2k+2,k],{k,0,n}],{n,0,20}]
%t (* or *)
%t Table[SeriesCoefficient[1/2*(2*x+(1-4*x)^(1/2)-1)/(1-4*x)^(1/2)/x^2/(-1+x),{x,0,n}],{n,0,20}] (* _Vaclav Kotesovec_, Oct 11 2012 *)
%t Table[(CatalanNumber[n + 1] (4 n + 6 - (n + 2) Hypergeometric2F1[1, -n-1, -n-1/2, 1/4]) - 1)/2, {n, 0, 20}] (* _Vladimir Reshetnikov_, Oct 03 2016 *)
%o (PARI) a(n) = sum(k=0, n, binomial(2*k+2, k)); \\ _Michel Marcus_, Oct 04 2016
%Y Cf. A000108, A001791.
%K nonn,easy,changed
%O 0,2
%A _Clark Kimberling_, Sep 07 2000
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