The OEIS is supported by the many generous donors to the OEIS Foundation.

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 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 (list; graph; refs; listen; history; text; internal format)
 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 Alois P. Heinz, Rows n = 0..200, flattened 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 Cf. A105633, A243752. Sequence in context: A344363 A163240 A091958 * A135306 A242352 A270953 Adjacent sequences: A116421 A116422 A116423 * A116425 A116426 A116427 KEYWORD nonn,tabf AUTHOR I. Tasoulas (jtas(AT)unipi.gr), Feb 15 2006 STATUS approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified June 10 02:46 EDT 2023. Contains 363183 sequences. (Running on oeis4.)