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A334648 a(n) is the total number of down steps between the second and third up steps in all 3_1-Dyck paths of length 4*n.

%I A334648 #15 Aug 08 2020 01:39:51
%S A334648 0,0,34,132,722,4638,32416,238956,1827918,14370595,115384756,
%T A334648 942115942,7798224226,65286060253,551838621972,4702955036640,
%U A334648 40366238473530,348631520142879,3027590307082804,26420699531880832,231571468023697960,2037650653547067005
%N A334648 a(n) is the total number of down steps between the second and third up steps in all 3_1-Dyck paths of length 4*n.
%C A334648 A 3_1-Dyck path is a lattice path with steps (1, 3), (1, -1) that starts and ends at y = 0 and stays above the line y = -1.
%C A334648 For n = 2, there is no 3rd up step, a(2) = 34 enumerates the total number of down steps between the 2nd up step and the end of the path.
%H A334648 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 A334648 a(0) = a(1) = 0 and a(n) = binomial(4*n+1, n)/(4*n+1) + 6*Sum_{j=1..2} binomial(4*j+2, j)*binomial(4*(n-j), n-j)/((4*j+2)*(n-j+1)) - 9*[n=2] for n > 1, where [ ] is the Iverson bracket.
%e A334648 For n = 2, the 3_1-Dyck paths are DUDDDUDD, DUDDUDDD, DUDUDDDD, DUUDDDDD, UDDDDUDD, UDDDUDDD, UDDUDDDD, UDUDDDDD, UUDDDDDD. In total, there are a(2) = 2 + 3 + 4 + 5 + 2 + 3 + 4 + 5 + 6 = 34 down steps between the 2nd up step and the end of the path.
%t A334648 a[0] = a[1] = 0; a[n_] := Binomial[4*n + 1, n]/(4*n + 1) + 6 * Sum[Binomial[4*j + 2, j] * Binomial[4*(n - j), n - j]/((4*j + 2)*(n - j + 1)), {j, 1, 2}] - 9 * Boole[n == 2]; Array[a, 22, 0] (* _Amiram Eldar_, May 12 2020 *)
%o A334648 (SageMath) [binomial(4*n + 1, n)/(4*n + 1) + 6*sum([binomial(4*j + 2, j)*binomial(4*(n - j), n - j)/(4*j + 2)/(n - j + 1) for j in srange(1, 3)]) - 9*(n==2) if n > 1 else 0 for n in srange(30)] # _Benjamin Hackl_, May 12 2020
%Y A334648 Cf. A002293, A007226, A007228, A334645, A334646, A334647, A334649, A334680, A334682, A334785.
%K A334648 nonn,easy
%O A334648 0,3
%A A334648 _Benjamin Hackl_, May 12 2020