A026781 a(n) = T(2n,n), T given by A026780.

1, 3, 12, 53, 246, 1178, 5768, 28731, 145108, 741392, 3825418, 19907156, 104370554, 550816506, 2924018194, 15603778253, 83661779470, 450479003038, 2435009205992, 13208558795146, 71879906857596, 392320357251928, 2147102400154768, 11780181236675858, 64782405317073968, 357022158144941548

Offset

0,2

Comments

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.

Links

G. C. Greubel, Table of n, a(n) for n = 0..1000

M. A. Alekseyev. On Enumeration of Dyck-Schroeder Paths. Journal of Combinatorial Mathematics and Combinatorial Computing 106 (2018), 59-68; arXiv:1601.06158 [math.CO], 2016-2018.

Formula

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

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

Maple

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

Mathematica

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 *)

Prog

(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 */
(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
(Sage)
def A026781_list(prec):
    P.<x> = PowerSeriesRing(ZZ, prec)
    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()
A026781_list(30) # G. C. Greubel, Nov 02 2019

Crossrefs

Cf. A026671.

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

Sequence in context: A151202 A151203 A262442 * A110122 A307412 A302188

Adjacent sequences: A026778 A026779 A026780 * A026782 A026783 A026784

Keyword

nonn

Author

Clark Kimberling

Extensions

More terms from Max Alekseyev, Jan 13 2015

Status

approved