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A334682 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).

%I A334682 #39 Apr 12 2025 06:48:52
%S A334682 0,3,18,118,829,6115,46736,366912,2941528,23981628,198224910,
%T A334682 1657364566,13992405626,119118427610,1021399476720,8813544248100,
%U A334682 76475285228304,666865500290884,5840843616021192,51361847992315320,453282040123194425,4013440075484640675
%N A334682 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 A334682 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 A334682 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 A334682 a(n) = binomial(4*(n+1)+1, n+1)/(4*(n+1)+1) - binomial(4*n+1, n)/(4*n+1).
%F A334682 a(n) = A062750(n+1, 3*n-1).
%F A334682 From _Muhammed Sefa Saydam_, Mar 01 2025: (Start)
%F A334682 Let F(n,k) = binomial((k+1)*n,n)/(k*n+1), then a(n) = F(n+1,3) - F(n,3).
%F A334682 Generally, F(y+1,x) - F(y,x) = Sum_{k=1..y} ( F(k+1,x) - T(x,k) + F(y-k,x) ) where T(n,k) = 2*(n+1)*binomial(n*k+k-1, k-1)/(n*k+1). (End)
%e A334682 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 A334682 b:= proc(x, y) option remember; `if`(x=y, x,
%p A334682      `if`(y+3<x, b(x-1, y+3), 0)+`if`(y>0, b(x-1, y-1), 0))
%p A334682     end:
%p A334682 a:= n-> b(4*n, 0):
%p A334682 seq(a(n), n=0..21);  # _Alois P. Heinz_, May 09 2020
%p A334682 # second Maple program:
%p A334682 a:= proc(n) option remember; `if`(n<2, 3*n, (8*(4*n-1)*
%p A334682       (2*n-1)*(4*n-3)*n*(229*n^2+303*n+98)*a(n-1))/
%p A334682       (3*(n-1)*(3*n+2)*(3*n+4)*(n+1)*(229*n^2-155*n+24)))
%p A334682     end:
%p A334682 seq(a(n), n=0..21);  # _Alois P. Heinz_, May 09 2020
%t A334682 nmax = 21;
%t A334682 A[_] = 0;
%t A334682 Do[A[x_] = 1 + x A[x]^4 + O[x]^(nmax + 2), nmax + 2];
%t A334682 CoefficientList[A[x], x] // Differences (* _Jean-François Alcover_, Aug 17 2020 *)
%o A334682 (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 A334682 First order differences of A002293.
%Y A334682 Cf. A062750.
%K A334682 nonn
%O A334682 0,2
%A A334682 _Andrei Asinowski_, May 08 2020