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A334719 a(n) is the total number of down-steps after the final up-step in all 4-Dyck paths of length 5*n (n up-steps and 4*n down-steps).

%I A334719 #33 Apr 28 2023 11:22:23
%S A334719 0,4,30,250,2245,21221,208129,2098565,21619910,226593015,2408424760,
%T A334719 25899375645,281273231985,3080585212120,33986840371400,
%U A334719 377364606387005,4213620859310140,47284625533425750,532996618440511710,6032169040263819485,68517222947120776290
%N A334719 a(n) is the total number of down-steps after the final up-step in all 4-Dyck paths of length 5*n (n up-steps and 4*n down-steps).
%C A334719 A 4-Dyck path is a lattice path with steps U = (1, 4), d = (1, -1) that starts at (0,0), stays (weakly) above the x-axis, and ends at the x-axis.
%H A334719 Stefano Spezia, <a href="/A334719/b334719.txt">Table of n, a(n) for n = 0..900</a>
%H A334719 Andrei Asinowski, Benjamin Hackl, and 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 A334719 a(n) = binomial(5*(n+1)+1, n+1)/(5*(n+1)+1) - binomial(5*n+1, n)/(5*n+1).
%F A334719 a(n) = A062985(n+1, 4*n-1).
%F A334719 G.f.: ((1 - x)*HypergeometricPFQ([1/5, 2/5, 3/5, 4/5], [1/2, 3/4, 5/4], 3125*x/256) - 1)/x. - _Stefano Spezia_, Apr 25 2023
%e A334719 For n = 2, the a(2) = 30 is the total number of down-steps after the last up-step in UddddUdddd, UdddUddddd, UddUdddddd, UdUddddddd, UUdddddddd (thus, 4 + 5 + 6 + 7 + 8).
%p A334719 b:= proc(x, y) option remember; `if`(x=y, x,
%p A334719      `if`(y+4<x, b(x-1, y+4), 0)+`if`(y>0, b(x-1, y-1), 0))
%p A334719     end:
%p A334719 a:= n-> b(5*n, 0):
%p A334719 seq(a(n), n=0..20);  # _Alois P. Heinz_, May 09 2020
%p A334719 # second Maple program:
%p A334719 a:= proc(n) option remember; `if`(n<2, 4*n, (5*(5*n-4)*
%p A334719       (5*n-3)*(5*n-2)*(5*n-1)*n*(2869*n^3+5354*n^2+3239*n+634)*
%p A334719        a(n-1))/(8*(n-1)*(4*n+3)*(2*n+1)*(4*n+5)*(n+1)*
%p A334719        (2869*n^3-3253*n^2+1138*n-120)))
%p A334719     end:
%p A334719 seq(a(n), n=0..20);  # _Alois P. Heinz_, May 09 2020
%t A334719 a[n_] := Binomial[5*n + 6, n + 1]/(5*n + 6) - Binomial[5*n + 1, n]/(5*n + 1); Array[a, 21, 0] (* _Amiram Eldar_, May 13 2020 *)
%o A334719 (PARI) a(n) = {binomial(5*(n+1)+1, n+1)/(5*(n+1)+1) - binomial(5*n+1, n)/(5*n+1)} \\ _Andrew Howroyd_, May 08 2020
%o A334719 (SageMath) [binomial(5*(n + 1) + 1, n + 1)/(5*(n + 1) + 1) - binomial(5*n + 1, n)/(5*n + 1) for n in srange(30)] # _Benjamin Hackl_, May 13 2020
%Y A334719 First order differences of A002294. Cf. A062985.
%Y A334719 Cf. A334682 (similar for 3-Dyck paths).
%K A334719 nonn,easy
%O A334719 0,2
%A A334719 _Andrei Asinowski_, May 08 2020