A359311 Number of Catalan paths (nonnegative, starting and ending at 0, step +/-1) of 2*n steps which reach at least 6 at some point.

0, 0, 0, 0, 0, 0, 1, 12, 89, 528, 2755, 13244, 60214, 263121, 1116791, 4637476, 18936940, 76327705, 304520286, 1205152900, 4738962369, 18540020091, 72240167011, 280579954028, 1087033982059, 4203231136230, 16228518078010, 62588797371361, 241198478726775

Offset: 0

Author

Greg Dresden, Jan 21 2023

Keywords

• nonn
• easy

Comments

a(n) = A000108(n) - A080937(n), which is #(Catalan paths) - #(Catalan paths of height <= 5).

Examples

			a(n) = 0 for n <= 5 because no path of length <= 10 can reach 6 and then descend to 0.
a(6) = 1 because there is one path of length 12 that reaches 6: six steps up, and six steps back down.
		

Crossrefs

Cf. A000108, A080936, A080937, A289419.

Programs

Maple

a:= n-> binomial(2*n, n)/(n+1)-(<<0|1|0>,
        <0|0|1>, <1|-6|5>>^n. <<1, 1, 2>>)[1, 1]:
seq(a(n), n=0..35);  # Alois P. Heinz, Jan 21 2023

Mathematica

Table[Sum[Binomial[2(n + 1), (n + 1) + 7 k] - 4 Binomial[2n, n + 7k], {k,1,n}], {n,0,30}]

Formula

a(n) = Sum_{k >= 1} binomial(2*(n+1), (n+1) + 7*k) - 4*binomial(2*n, n+7*k).

From Alois P. Heinz, Jan 21 2023: (Start)

G.f.: (1-sqrt(1-4*x))/(2*x) - (1-4*x+3*x^2)/(1-5*x+6*x^2-x^3).

a(n) = Sum_{k=6..n} A080936(n,k). (End)

D-finite with recurrence -(n+1)*(n-6)*a(n) +3*(3*n^2-17*n+4)*a(n-1) +2*(-13*n^2+80*n-87)*a(n-2) +(25*n^2-161*n+246)*a(n-3) -2*(n-3)*(2*n-7)*a(n-4)=0. - R. J. Mathar, Jan 25 2023