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A334640 a(n) is the total number of down steps between the 2nd and 3rd up steps in all 2-Dyck paths of length 3*n. A 2-Dyck path is a nonnegative lattice path with steps (1, 2), (1, -1) that starts and ends at y = 0.

%I A334640 #30 Aug 08 2020 01:42:37
%S A334640 0,0,9,19,72,324,1595,8307,44982,250648,1427679,8274825,48644310,
%T A334640 289334160,1738043892,10529070020,64252519830,394601627376,
%U A334640 2437058926871,15126463230165,94306717535940,590318477063700,3708527622652755,23374587898663155,147770791807427880
%N A334640 a(n) is the total number of down steps between the 2nd and 3rd up steps in all 2-Dyck paths of length 3*n. A 2-Dyck path is a nonnegative lattice path with steps (1, 2), (1, -1) that starts and ends at y = 0.
%C A334640 For n = 2, there is no 3rd up step, a(2) = 9 enumerates the total number of down steps between the 2nd up step and the end of the path.
%H A334640 Alois P. Heinz, <a href="/A334640/b334640.txt">Table of n, a(n) for n = 0..1212</a>
%H A334640 A. Asinowski, B. Hackl, and S. Selkirk, <a href="https://arxiv.org/abs/2007.15562">Down step statistics in generalized Dyck paths</a>, arXiv:2007.15562 [math.CO], 2020.
%F A334640 a(0) = a(1) = 0 and a(n) = 2*Sum_{j=1..2} binomial(3*j+1,j) * binomial(3*(n-j),n-j) / ((3*j+1)*(n-j+1)) for n > 1.
%e A334640 For n = 2, there are the 2-Dyck paths UUDDDD, UDUDDD, UDDUDD. Between the 2nd up step and the end of the path there are a(2) = 4 + 3 + 2 = 9 down steps in total.
%p A334640 b:= proc(x, y, u, c) option remember; `if`(x=0, c,
%p A334640      `if`(y+2<x, b(x-1, y+2, min(u+1,3), c), 0)+
%p A334640      `if`(y>0, b(x-1, y-1, u, c+`if`(u=2, 1, 0)), 0))
%p A334640     end:
%p A334640 a:= n-> b(3*n, 0$3):
%p A334640 seq(a(n), n=0..24);  # _Alois P. Heinz_, May 09 2020
%p A334640 # second Maple program:
%p A334640 a:= proc(n) option remember; `if`(n<3, [0$2, 9][n+1],
%p A334640      (3*(n-1)*(3*n-8)*(3*n-7)*(13*n-20)*a(n-1))/
%p A334640      (2*(13*n-33)*(n-2)*(2*n-3)*n))
%p A334640     end:
%p A334640 seq(a(n), n=0..24);  # _Alois P. Heinz_, May 09 2020
%t A334640 a[0] = a[1] = 0; a[n_] := 2 * Sum[Binomial[3*j + 1, j] * Binomial[3*(n - j), n - j]/((3*j + 1)*(n - j + 1)), {j, 1, 2}]; Array[a, 25, 0] (* _Amiram Eldar_, May 09 2020 *)
%o A334640 (PARI) a(n) = if (n<=1, 0, 2*sum(j=1, 2, binomial(3*j+1,j) * binomial(3*(n-j),n-j)/((3*j+1)*(n-j+1)))); \\ _Michel Marcus_, May 09 2020
%Y A334640 Cf. A007226, A007228, A124724, A334641, A334642, A334682.
%K A334640 nonn,easy
%O A334640 0,3
%A A334640 _Benjamin Hackl_, May 07 2020