A116424 Triangle read by rows: T(n,k) = the number of Dyck paths of semilength n with k UDUU's, 0 <= k <= floor((n-1)/2).
1, 1, 2, 4, 1, 9, 5, 22, 19, 1, 57, 66, 9, 154, 221, 53, 1, 429, 729, 258, 14, 1223, 2391, 1131, 116, 1, 3550, 7829, 4652, 745, 20, 10455, 25638, 18357, 4115, 220, 1, 31160, 84033, 70404, 20598, 1790, 27, 93802, 275765, 264563, 96286, 12104, 379, 1, 284789
Offset: 0
Examples
Triangle begins: 00 : 1; 01 : 1; 02 : 2; 03 : 4, 1; 04 : 9, 5; 05 : 22, 19, 1; 06 : 57, 66, 9; 07 : 154, 221, 53, 1; 08 : 429, 729, 258, 14; 09 : 1223, 2391, 1131, 116, 1; 10 : 3550, 7829, 4652, 745, 20; ... T(4,1) = 5 because there exist five Dyck paths of semilength 4 with one occurrence of UDUU : UDUUUDDD, UDUUDUDD, UDUUDDUD, UUDUUDDD, UDUDUUDD.
Links
- Alois P. Heinz, Rows n = 0..200, flattened
- Jean-Luc Baril, Pamela E. Harris, Kimberly J. Harry, Matt McClinton, and José L. Ramírez, Enumerating runs, valleys, and peaks in Catalan words, arXiv:2404.05672 [math.CO], 2024. See p. 18.
- Toufik Mansour, Statistics on Dyck Paths, Journal of Integer Sequences, Vol. 9 (2006), Article 06.1.5.
- A. Sapounakis, I. Tasoulas and P. Tsikouras, Counting strings in Dyck paths, Discrete Math., 307 (2007), 2909-2924.
Programs
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Maple
b:= proc(x, y, t) option remember; `if`(y<0 or y>x, 0, `if`(x=0, 1, expand(b(x-1, y+1, [2, 2, 4, 2][t])* `if`(t=4, z, 1) +b(x-1, y-1, [1, 3, 1, 3][t])))) end: T:= n-> (p-> seq(coeff(p, z, i), i=0..degree(p)))(b(2*n, 0, 1)): seq(T(n), n=0..15); # Alois P. Heinz, Jun 02 2014 -
Mathematica
s = Series[((1 + (t - 1) z^2) - Sqrt[(1 + (t - 1) z^2)^2 - 4*z*(1 - z + z*t)])/(2*z*(1 - z + z*t)), {z, 0, 15}] // CoefficientList[#, z]&; CoefficientList[#, t]& /@ s // Flatten (* updated by Jean-François Alcover, Feb 14 2021 *)
Formula
T(n,k) = Sum_{i=k..floor((n-1)/2)} (-1)^(i+k) * binomial(i,k) * binomial(n-i,i) * binomial(2*n-3*i, n - 2*i -1)/(n-i), n >= 1.
G.f. G = G(t,z) satisfies G = 1 + z^2(1-t)G + z(1-z+tz)G^2.
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