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%I #21 Apr 28 2023 11:22:05
%S 0,9,82,747,7065,69098,694272,7127865,74468546,789265125,8466019380,
%T 91736269053,1002710879409,11042713886256,122413333216960,
%U 1364880618458565,15296452128008100,172218124701600741,1946960139291303222,22092883135853433030,251545025683283255770
%N a(n) is the total number of down-steps after the final up-step in all 4_2-Dyck paths of length 5*n (n up-steps and 4*n down-steps).
%C A 4_2-Dyck path is a lattice path with steps U = (1, 4), d = (1, -1) that starts at (0,0), stays (weakly) above y = -2, and ends at the x-axis.
%H Stefano Spezia, <a href="/A334611/b334611.txt">Table of n, a(n) for n = 0..900</a>
%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(5*(n+1)+3, n+1)/(5*(n+1)+3) - 9*binomial(5*n+3, n)/(5*n+3).
%F G.f.: ((1 - 3*x)*HypergeometricPFQ([3/5, 4/5, 6/5, 7/5], [5/4, 3/2, 7/4], 3125*x/256) - 1)/x. - _Stefano Spezia_, Apr 25 2023
%e For n=1, a(1) = 9 is the total number of down-steps after the last up-step in Udddd, dUddd, ddUdd.
%t a[n_] := 3 * Binomial[5*n + 8, n + 1]/(5*n + 8) - 9 * Binomial[5*n + 3, n]/(5*n + 3); Array[a, 21, 0] (* _Amiram Eldar_, May 13 2020 *)
%Y Cf. A334786, A334719, A334610, A334612.
%K nonn
%O 0,2
%A _Andrei Asinowski_, May 13 2020