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a(n) is the total number of down-steps after the final up-step in all 3-Dyck paths of length 4*n (n up-steps and 3*n down-steps).
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%I #27 Aug 17 2020 07:56:53

%S 0,3,18,118,829,6115,46736,366912,2941528,23981628,198224910,

%T 1657364566,13992405626,119118427610,1021399476720,8813544248100,

%U 76475285228304,666865500290884,5840843616021192,51361847992315320,453282040123194425,4013440075484640675

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

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

%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(n) = binomial(4*(n+1)+1, n+1)/(4*(n+1)+1) - binomial(4*n+1, n)/(4*n+1).

%F a(n) = A062750(n+1, 3*n-1).

%e For n=2 the a(2)=18 is the total number of down-steps after the last up-step in UdddUddd, UddUdddd, UdUddddd, UUdddddd.

%p b:= proc(x, y) option remember; `if`(x=y, x,

%p `if`(y+3<x, b(x-1, y+3), 0)+`if`(y>0, b(x-1, y-1), 0))

%p end:

%p a:= n-> b(4*n, 0):

%p seq(a(n), n=0..21); # _Alois P. Heinz_, May 09 2020

%p # second Maple program:

%p a:= proc(n) option remember; `if`(n<2, 3*n, (8*(4*n-1)*

%p (2*n-1)*(4*n-3)*n*(229*n^2+303*n+98)*a(n-1))/

%p (3*(n-1)*(3*n+2)*(3*n+4)*(n+1)*(229*n^2-155*n+24)))

%p end:

%p seq(a(n), n=0..21); # _Alois P. Heinz_, May 09 2020

%t nmax = 21;

%t A[_] = 0;

%t Do[A[x_] = 1 + x A[x]^4 + O[x]^(nmax + 2), nmax + 2];

%t CoefficientList[A[x], x] // Differences (* _Jean-François Alcover_, Aug 17 2020 *)

%o (PARI) a(n) = {binomial(4*(n+1)+1, n+1)/(4*(n+1)+1) - binomial(4*n+1, n)/(4*n+1)} \\ _Andrew Howroyd_, May 08 2020

%Y First order differences of A002293.

%Y Cf. A062750.

%K nonn

%O 0,2

%A _Andrei Asinowski_, May 08 2020