A124724 a(n) = (4/(n + 1)) * C(5*n, n).

4, 10, 60, 455, 3876, 35420, 339300, 3362260, 34179860, 354465254, 3735373880, 39884521950, 430571952300, 4691735290080, 51534335175776, 570003171679020, 6343110854237300, 70968228417131850, 797820661622862900

Offset: 0

Author

Paul Barry, Nov 05 2006

Keywords

• easy
• nonn

Comments

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

Links

• Michael De Vlieger, Table of n, a(n) for n = 0..924
• A. Asinowski, B. Hackl, and S. Selkirk, Down step statistics in generalized Dyck paths, arXiv:2007.15562 [math.CO], 2020.
• Fabio Deelan Cunden, Marilena Ligabò, and Tommaso Monni, Random matrices associated to Young diagrams, arXiv:2301.13555 [math.PR], 2023. See p. 7.
• N. S. S. Gu, H. Prodinger, and S. Wagner, Bijections for a class of labeled plane trees, Eur. J. Combinat. 31 (2010) 720-732; see Theorem 2 with k = 4.

Crossrefs

Cf. A007226, A007228.

Programs

Mathematica

Array[(4/(# + 1))*Binomial[5 #, #] &, 28, 0] (* Michael De Vlieger, Apr 12 2023 *)

PARI

a(n) = (4/(n+1)) * binomial(5*n,n); \\ Michel Marcus, May 08 2020

Formula

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).