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%I #14 Mar 26 2022 07:47:41
%S 0,0,3,19,108,609,3468,20007,116886,690690,4122495,24823188,150629248,
%T 920274804,5656456104,34954487967,217044280458,1353539406660,
%U 8474029162305,53241343026795,335592121524660,2121577490385885,13448859209014320,85467026778421860
%N a(n) is the total number of down steps between the (n-1)-th and n-th up steps in all 2-Dyck paths of length 3*n. A 2-Dyck path is a nonnegative path with steps (1, 2), (1, -1) that starts and ends at y = 0.
%C For n = 1, there is no (n-1)-th up step, a(1) = 0 is the total number of down steps before the first up step.
%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) = 0 and a(n) = binomial(3*n+4, n+1)/(3*n+4) - 3*binomial(3*n+1, n)/(3*n+1) for n > 0.
%e For n = 2, the 2-Dyck paths are UDDUDD, UDUDDD, UUDDDD. Therefore the total number of down steps between the first and second up steps is a(2) = 2+1+0 = 3.
%p alias(PS=ListTools:-PartialSums): A334976List := proc(m) local A, P, n;
%p A := [0,0]; P := [1,0]; for n from 1 to m - 2 do P := PS(PS([op(P), P[-1]]));
%p A := [op(A), P[-1]] od; A end: A334976List(24); # _Peter Luschny_, Mar 26 2022
%t a[0] = 0; a[n_] := Binomial[3*n+4, n+1]/(3*n + 4) - 3 * Binomial[3*n + 1, n]/(3*n + 1); Array[a, 24, 0]
%o (SageMath) [binomial(3*n + 4, n + 1)/(3*n + 4) - 3*binomial(3*n + 1, n)/(3*n + 1) if n > 0 else 0 for n in srange(30)] # _Benjamin Hackl_, May 19 2020
%Y Cf. A334977, A334978, A334979, A334980.
%K nonn,easy
%O 0,3
%A _Sarah Selkirk_, May 18 2020
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