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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).
3
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
0,3
COMMENTS
T(n,k) also gives the number of Dyck paths of semilength n with k UUDU's.
Column k=0 gives A105633(n-1) for n > 0.
LINKS
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.
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.
EXAMPLE
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.
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 *)
CROSSREFS
Sequence in context: A163240 A091958 A372879 * A372883 A135306 A242352
KEYWORD
nonn,tabf
AUTHOR
I. Tasoulas (jtas(AT)unipi.gr), Feb 15 2006
STATUS
approved