|
%I #13 Aug 08 2020 01:39:37
%S 0,0,0,118,409,2368,15750,112716,845295,6551208,52035714,421286280,
%T 3463401007,28832656408,242565115858,2058945519936,17611312647075,
%U 151647023490480,1313460091978458,11435310622320552,100019000856225156,878443730199290560
%N a(n) is the total number of down steps between the 3rd and 4th up steps in all 3-Dyck paths of length 4*n.
%C A 3-Dyck path is a nonnegative lattice path with steps (1, 3), (1, -1) that starts and ends at y = 0.
%H Andrei Asinowski, Benjamin Hackl, Sarah J. Selkirk, <a href="https://arxiv.org/abs/2007.15562">Down-step statistics in generalized Dyck paths</a>, arXiv:2007.15562 [math.CO], 2020.
%F a(0) = a(1) = a(2) = 0 and a(n) = 3*Sum_{j=1..3} binomial(4*j+1, j)*binomial(4*(n-j), n-j)/((4*j+1)*(n-j+1)) for n > 2.
%e For n = 3, there is no 4th up step, a(3) = 118 enumerates the total number of down steps between the 3rd up step and the end of the path.
%t a[0] = a[1] = a[2] = 0; a[n_] := 3 * Sum[Binomial[4*j + 1, j] * Binomial[4*(n - j), n - j]/((4*j + 1)*(n - j + 1)), {j, 1, 3}]; Array[a, 22, 0] (* _Amiram Eldar_, May 12 2020 *)
%o (SageMath) [3*sum([binomial(4*j + 1, j)*binomial(4*(n - j), n - j)/(4*j + 1)/(n - j + 1) for j in srange(1, 4)]) if n > 2 else 0 for n in srange(30)] # _Benjamin Hackl_, May 12 2020
%Y Cf. A002293, A007226, A007228, A334641, A334645, A334682, A334785.
%K nonn,easy
%O 0,4
%A _Benjamin Hackl_, May 12 2020
|
