A334609 - OEIS

%I #15 Aug 07 2020 12:08:26

%S 0,6,46,339,2553,19723,155805,1253931,10249096,84864051,710429304,

%T 6003238901,51140131770,438729741450,3787208722815,32871470376123,

%U 286706337100656,2511620756461504,22089299382478728,194966351598215340,1726424465382128205

%N a(n) is the total number of down-steps after the final up-step in all 3_2-Dyck paths of length 4*n (n up-steps and 3*n down-steps).

%C A 3_2-Dyck path is a lattice path with steps U = (1, 3), d = (1, -1) that starts at (0,0), stays (weakly) above y = -2, and ends at the x-axis.

%H 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 a(n) = 3*binomial(4*(n+1) + 3, n+1)/(4*(n+1) + 3) - 9*binomial(4*n+3, n)/(4*n + 3).

%e For n = 1, a(1) = 6 is the total number of down-steps after the last up-step in Uddd, dUdd, ddUd.

%t a[n_] := 3 * Binomial[4*n + 7, n + 1]/(4*n + 7) - 9 * Binomial[4*n + 3, n]/(4*n + 3); Array[a, 21, 0] (* _Amiram Eldar_, May 13 2020 *)

%o (SageMath) [3*binomial(4*(n + 1) + 3, n + 1)/(4*(n + 1) + 3) - 9*binomial(4*n + 3, n)/(4*n + 3) for n in srange(30)] # _Benjamin Hackl_, May 13 2020

%Y Cf. A334785, A334650, A334682, A334608.

%K nonn,easy

%O 0,2

%A _Andrei Asinowski_, May 13 2020