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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.

%I A359311 #124 Jan 25 2023 09:09:26
%S A359311 0,0,0,0,0,0,1,12,89,528,2755,13244,60214,263121,1116791,4637476,
%T A359311 18936940,76327705,304520286,1205152900,4738962369,18540020091,
%U A359311 72240167011,280579954028,1087033982059,4203231136230,16228518078010,62588797371361,241198478726775
%N 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.
%C A359311 a(n) = A000108(n) - A080937(n), which is #(Catalan paths) - #(Catalan paths of height <= 5).
%F A359311 a(n) = Sum_{k >= 1} binomial(2*(n+1), (n+1) + 7*k) - 4*binomial(2*n, n+7*k).
%F A359311 From _Alois P. Heinz_, Jan 21 2023: (Start)
%F A359311 G.f.: (1-sqrt(1-4*x))/(2*x) - (1-4*x+3*x^2)/(1-5*x+6*x^2-x^3).
%F A359311 a(n) = Sum_{k=6..n} A080936(n,k). (End)
%F A359311 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
%e A359311 a(n) = 0 for n <= 5 because no path of length <= 10 can reach 6 and then descend to 0.
%e A359311 a(6) = 1 because there is one path of length 12 that reaches 6: six steps up, and six steps back down.
%p A359311 a:= n-> binomial(2*n, n)/(n+1)-(<<0|1|0>,
%p A359311         <0|0|1>, <1|-6|5>>^n. <<1, 1, 2>>)[1, 1]:
%p A359311 seq(a(n), n=0..35);  # _Alois P. Heinz_, Jan 21 2023
%t A359311 Table[Sum[Binomial[2(n + 1), (n + 1) + 7 k] - 4 Binomial[2n, n + 7k], {k,1,n}], {n,0,30}]
%Y A359311 Cf. A000108, A080936, A080937, A289419.
%K A359311 nonn,easy
%O A359311 0,8
%A A359311 _Greg Dresden_, Jan 21 2023