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%I #35 Nov 29 2022 01:21:44
%S 1,1,3,7,19,51,143,407,1183,3487,10415,31439,95791,294191,909823,
%T 2830943,8856255,27839167,87888767,278545663,885903743,2826612095,
%U 9045147391,29022168063,93350430975,300949170431,972271227647
%N Expansion of g.f.: (1-2*x-sqrt(1-4*x+8*x^3-4*x^4))/(2*x^2*(1-x)).
%C Sequence is the binomial transform of the aerated large Schroeder numbers A006318. Hankel transform is A060656(n+1).
%C Number of lattice paths, never going below the x-axis, from (0,0) to (n,0) consisting of up steps U = (1,1), down steps D = (1,-1) and horizontal steps H(k) = (k,0) for every positive integer k. For instance, for n=3, we have the 7 paths: H(1)H(1)H(1), H(1)H(2), H(2)H(1), H(3), H(1)UD, UDH(1), UH(1)D. - _Emanuele Munarini_, Mar 14 2011
%H Alois P. Heinz, <a href="/A135052/b135052.txt">Table of n, a(n) for n = 0..1000</a>
%H Paul Barry, <a href="https://arxiv.org/abs/2211.12637">Conjectures on Somos 4, 6 and 8 sequences using Riordan arrays and the Catalan numbers</a>, arXiv:2211.12637 [math.CO], 2022.
%H R. De Castro, A. L. Ramírez and J. L. Ramírez, <a href="http://arxiv.org/abs/1310.2449">Applications in Enumerative Combinatorics of Infinite Weighted Automata and Graphs</a>, arXiv:1310.2449 [cs.DM], 2013.
%H R. De Castro, A. L. Ramírez and J. L. Ramírez, <a href="http://dx.doi.org/10.7561/SACS.2014.1.137">Applications in Enumerative Combinatorics of Infinite Weighted Automata and Graphs</a>, Scientific Annals of Computer Science, 24(1)(2014), 137-171
%F a(n) = Sum_{k=0..n, Sum_{j=0..k/2, C(k/2+j, 2j)*C(j)*(1+(-1)^k)/2}}, where C(n) is A000108(n).
%F G.f.: 1/(1-x-2x^2/(1-x-x^2/(1-x-2x^2/(1-x-x^2/(1-x-2x^2.... (continued fraction). - _Paul Barry_, Jan 02 2009
%F a(n) = Sum_{s=0..n} Sum_{m=0..n-2s} (C(s)*binomial(m+2s,m) * binomial(n-2s-1,m-1)), where C(n) is A000108(n). - _José Luis Ramírez Ramírez_, Apr 19 2015
%F Conjecture: (n+2)*a(n) +(-5*n-4)*a(n-1) +2*(2*n+1)*a(n-2) +4*(2*n-5)*a(n-3) +12*(-n+3)*a(n-4) +4*(n-4)*a(n-5)=0. - _R. J. Mathar_, Apr 19 2015
%F a(n) ~ (2+sqrt(2))^(n+1) / (sqrt(Pi) * n^(3/2)). - _Vaclav Kotesovec_, Apr 20 2015
%t CoefficientList[Series[(1 - 2 x - Sqrt[1 - 4 x + 8 x^3 - 4 x^4]) / (2 x^2 (1 - x)), {x, 0, 33}], x] (* _Vincenzo Librandi_, Apr 19 2015 *)
%K easy,nonn
%O 0,3
%A _Paul Barry_, Nov 15 2007
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