%I #26 Jan 13 2024 13:07:18

%S 1,3,12,53,246,1178,5768,28731,145108,741392,3825418,19907156,

%T 104370554,550816506,2924018194,15603778253,83661779470,450479003038,

%U 2435009205992,13208558795146,71879906857596,392320357251928,2147102400154768,11780181236675858,64782405317073968,357022158144941548

%N a(n) = T(2n,n), T given by A026780.

%C Number of paths from (0,0) to (n,n) in the directed graph having vertices (i,j) and edges (i,j)-to-(i+1,j) and (i,j)-to-(i,j+1) for i,j>=0 and edges (i,i+h)-to-(i+1,i+h+1) for i>=0, h>=0.

%H G. C. Greubel, <a href="/A026781/b026781.txt">Table of n, a(n) for n = 0..1000</a>

%H M. A. Alekseyev. <a href="https://arxiv.org/abs/1601.06158">On Enumeration of Dyck-Schroeder Paths</a>. Journal of Combinatorial Mathematics and Combinatorial Computing 106 (2018), 59-68; arXiv:1601.06158 [math.CO], 2016-2018.

%F O.g.f.: S(x)/(1-x*C(x)*S(x)) = (S(x)-C(x))/(x*C(x)), where C(x)=(1-sqrt(1-4x))/(2*x) is o.g.f. for A000108 and S(x)=(1-x-sqrt(1-6*x+x^2))/(2*x) is o.g.f. for A006318. - _Max Alekseyev_, Jan 13 2015

%F D-finite with recurrence 2*n*(132*n-445)*(n+2)*(n+1)*a(n) -n*(n+1) *(5587*n^2 -23082*n +12800)*a(n-1) +2*n*(n-1)*(22870*n^2 -114505*n +116854)*a(n-2) +2*(-90081*n^4 +818062*n^3 -2626791*n^2 +3517598*n -1622544)*a(n-3) +4*(85519*n^4 -1071535*n^3 +4986308*n^2 -10177616*n +7647024)*a(n-4) +(-269235*n^4 +4490125*n^3 -27985152*n^2 +77217236*n -79534224)*a(n-5) +4*(2*n-11)*(8203*n^3 -117312*n^2 +557264*n -879984)*a(n-6) -4*(n-6)*(307*n -1414) *(2*n-11) *(2*n-13)*a(n-7)=0. - _R. J. Mathar_, Feb 20 2020

%p seq(coeff(series(2*(1-x -sqrt(1-6*x+x^2))/(4*x -(1 -sqrt(1-4*x))*(1 -x -sqrt(1-6*x+x^2))), x, n+1), x, n), n = 0..30); # _G. C. Greubel_, Nov 02 2019

%t CoefficientList[Series[2*(1-x -Sqrt[1-6*x+x^2])/(4*x -(1 -Sqrt[1-4*x])*(1 -x -Sqrt[1-6*x+x^2])), {x,0,30}], x] (* _G. C. Greubel_, Nov 02 2019 *)

%o (PARI) C = (1-sqrt(1-4*x+O(x^51)))/2/x; S = (1-x-sqrt(1-6*x+x^2 +O(x^51) ))/2/x; Vec(S/(1-x*C*S)) /* _Max Alekseyev_, Jan 13 2015 */

%o (Magma) R<x>:=PowerSeriesRing(Rationals(), 30); Coefficients(R!( 2*(1-x -Sqrt(1-6*x+x^2))/(4*x -(1 -Sqrt(1-4*x))*(1 -x -Sqrt(1-6*x+x^2))) )); // _G. C. Greubel_, Nov 02 2019

%o (Sage)

%o def A026781_list(prec):

%o P.<x> = PowerSeriesRing(ZZ, prec)

%o return P(2*(1-x -sqrt(1-6*x+x^2))/(4*x -(1 -sqrt(1-4*x))*(1 -x -sqrt(1-6*x+x^2)))).list()

%o A026781_list(30) # _G. C. Greubel_, Nov 02 2019

%Y Cf. A026671.

%Y Cf. A026780, A026782, A026783, A026784, A026785, A026786, A026787, A026788, A026789, A026790.

%K nonn

%O 0,2

%A _Clark Kimberling_

%E More terms from _Max Alekseyev_, Jan 13 2015