%I #28 May 02 2024 04:48:41

%S 1,4,13,40,120,356,1050,3088,9069,26620,78133,229384,673699,1979628,

%T 5820195,17121312,50394579,148413996,437324919,1289330520,3803175474,

%U 11223840012,33139076292,97889042384,289276841475,855205791076,2529279459099

%N a(n) = T(n,n+1), where T is the array defined in A025564.

%H Michael De Vlieger, <a href="/A025567/b025567.txt">Table of n, a(n) for n = 1..1000</a>

%H Jean-Luc Baril, Richard Genestier, Sergey Kirgizov, <a href="https://arxiv.org/abs/1911.03119">Pattern distributions in Dyck paths with a first return decomposition constrained by height</a>, arXiv:1911.03119 [math.CO], 2019.

%H Luca Ferrari and Emanuele Munarini, <a href="http://arxiv.org/abs/1203.6792">Enumeration of edges in some lattices of paths</a>, arXiv preprint arXiv:1203.6792 [math.CO], 2012 and <a href="https://cs.uwaterloo.ca/journals/JIS/VOL17/Ferrari/ferrari.html">J. Int. Seq. 17 (2014) #14.1.5</a>

%F G.f.: (x^2-1-sqrt(1+x)*(x^2+2*x-1)/sqrt(1-3*x))/(2*x^3). - _Mark van Hoeij_, May 01 2013

%F Conjecture: (n+3)*a(n) +4*(-n-2)*a(n-1) +2*a(n-2) +8*(n-1)*a(n-3) +3*(n-3)*a(n-4)=0. - _R. J. Mathar_, Apr 03 2015

%F Conjecture: (n-1)*(n-2)*(n+3)*a(n) -2*n*(n-2)*(n+2)*a(n-1) -3*n*(n-1)^2*a(n-2)=0. - _R. J. Mathar_, Apr 03 2015

%F a(n) ~ 2 * 3^(n + 1/2) / sqrt(Pi*n). - _Vaclav Kotesovec_, May 02 2024

%t T[_, 0] = 1; T[1, 1] = 2; T[n_, k_] /; 0 <= k <= 2n := T[n, k] = T[n-1, k-2] + T[n-1, k-1] + T[n-1, k]; T[_, _] = 0;

%t a[n_] := T[n+1, n+3];

%t Array[a, 27] (* _Jean-François Alcover_, Oct 30 2018 *)

%o (PARI) x='x+O('x^66); Vec((x^2-1-sqrt(1+x)*(x^2+2*x-1)/sqrt(1-3*x))/(2*x^3)) \\ _Joerg Arndt_, May 01 2013

%Y Pairwise sums of A014531.

%K nonn

%O 1,2

%A _Clark Kimberling_