|
%I #9 Aug 08 2020 01:39:46
%S 0,0,0,236,1034,6094,40996,295740,2231022,17370163,138473536,
%T 1124433142,9266859394,77307427741,651540030688,5538977450256,
%U 47442103851930,409000732566399,3546232676711824,30903652601552272,270529448396053576,2377829916885541565
%N a(n) is the total number of down steps between the third and fourth up steps in all 3_1-Dyck paths of length 4*n.
%C 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 For n = 3, there is no 4th up step, a(3) = 236 enumerates the total number of down steps between the 3rd up step and the end of the path.
%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) = binomial(4*n+1, n)/(4*n+1) + 6*Sum_{j=1..3} binomial(4*j+2, j)*binomial(4*(n-j), n-j)/((4*j+2)*(n-j+1)) - 52*[n=3] for n > 2, where [ ] is the Iverson bracket.
%o (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, 4)]) - 52*(n==3) if n > 2 else 0 for n in srange(30)] # _Benjamin Hackl_, May 12 2020
%Y Cf. A002293, A007226, A007228, A334645, A334646, A334647, A334648, A334680, A334682, A334785.
%K nonn,easy
%O 0,4
%A _Benjamin Hackl_, May 12 2020
|
