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%I #24 Apr 12 2023 11:40:01
%S 4,10,60,455,3876,35420,339300,3362260,34179860,354465254,3735373880,
%T 39884521950,430571952300,4691735290080,51534335175776,
%U 570003171679020,6343110854237300,70968228417131850,797820661622862900
%N a(n) = (4/(n + 1)) * C(5*n, n).
%C a(n) is the total number of down steps between the first and second up steps in all 4-Dyck paths of length 5*(n+1). A 4-Dyck path is a nonnegative lattice path with steps (1,4), (1,-1) that starts and ends at y = 0. - _Sarah Selkirk_, May 07 2020
%H Michael De Vlieger, <a href="/A124724/b124724.txt">Table of n, a(n) for n = 0..924</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.
%H Fabio Deelan Cunden, Marilena Ligabò, and Tommaso Monni, <a href="https://arxiv.org/abs/2301.13555">Random matrices associated to Young diagrams</a>, arXiv:2301.13555 [math.PR], 2023. See p. 7.
%H N. S. S. Gu, H. Prodinger, and S. Wagner, <a href="https://doi.org/10.1016/j.ejc.2009.10.007">Bijections for a class of labeled plane trees</a>, Eur. J. Combinat. 31 (2010) 720-732; see Theorem 2 with k = 4.
%F a(n) = C(5*n, n)/(4*n + 1) + 2*C(5*n + 1, n)/(4*n + 2) + 3*C(5*n + 2, n)/(4*n + 3) + 4*C(5*n + 3, n)/(4*n + 4).
%t Array[(4/(# + 1))*Binomial[5 #, #] &, 28, 0] (* _Michael De Vlieger_, Apr 12 2023 *)
%o (PARI) a(n) = (4/(n+1)) * binomial(5*n,n); \\ _Michel Marcus_, May 08 2020
%Y Cf. A007226, A007228.
%K easy,nonn
%O 0,1
%A _Paul Barry_, Nov 05 2006
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