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A026781
a(n) = T(2n,n), T given by A026780.
14
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
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
KEYWORD
nonn
EXTENSIONS
More terms from Max Alekseyev, Jan 13 2015
STATUS
approved