A114492 Triangle read by rows: T(n,k) is the number of Dyck paths of semilength n and having k DDUU's, where U=(1,1), D=(1,-1) (0<=k<=floor(n/2)-1 for n>=2).

1, 1, 2, 5, 13, 1, 35, 7, 97, 34, 1, 275, 143, 11, 794, 558, 77, 1, 2327, 2083, 436, 16, 6905, 7559, 2180, 151, 1, 20705, 26913, 10051, 1095, 22, 62642, 94547, 43796, 6758, 268, 1, 190987, 328943, 183130, 37402, 2409, 29, 586219, 1136218, 742253, 191408

Offset

0,3

Comments

Rows 0 and 1 contain one term each; row n contains floor(n/2) terms (n>=2).

Row sums are the Catalan numbers (A000108). Column 0 yields A086581.

Sum(k*T(n,k),k=0..floor(n/2)-1) = binomial(2n-3,n-4) (A003516).

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. 12.

FindStat - Combinatorial Statistic Finder, The number of occurrences of the contiguous pattern [.,[.,[[.,.],.]]].

Toufik Mansour and Mark Shattuck, Counting occurrences of subword patterns in non-crossing partitions, Art Disc. Appl. Math. (2022).

A. Sapounakis, I. Tasoulas and P. Tsikouras, Counting strings in Dyck paths, Discrete Math., 307 (2007), 2909-2924.

Formula

G.f.: G=G(t, z) satisfies z(t+z-tz)G^2-(1-2(1-t)z+(1-t)z^2)G+1-z+tz=0.

Example

T(5,1) = 7 because we have UU(DDUU)DUDD, UU(DDUU)UDDD, UDUU(DDUU)DD, their mirror images and UUU(DDUU)DDD (the DDUU's are shown between parentheses).
Triangle starts:
   1;
   1;
   2;
   5;
  13,  1;
  35,  7;
  97, 34, 1;
  ...

Maple

G:=1/2/(-t*z-z^2+z^2*t)*(-1+2*z-2*t*z-z^2+z^2*t+sqrt(1+z^4-2*z^4*t+z^4*t^2-4*z+2*z^2-2*z^2*t)): Gser:=simplify(series(G, z=0, 17)): P[0]:=1: for n from 1 to 14 do P[n]:=coeff(Gser, z^n) od: 1; 1; for n from 0 to 14 do seq(coeff(t*P[n], t^j), j=1..floor(n/2)) od; # yields sequence in triangular form

Mathematica

m = 15; G[_, _] = 0;
Do[G[t_, z_] = (-1 + z - t z - t z G[t, z]^2 - z^2 G[t, z]^2 + t z^2 G[t, z]^2)/(-1 + 2z - 2t z - z^2 + t z^2) + O[t]^Floor[m/2] + O[z]^m, {m}];
CoefficientList[#, t]& /@ Take[CoefficientList[G[t, z], z], m] // Flatten (* Jean-François Alcover, Oct 05 2019 *)

Crossrefs

Cf. A000108, A086581, A003516.

Sequence in context: A372875 A114502 A135308 * A135305 A114463 A135309

Adjacent sequences: A114489 A114490 A114491 * A114493 A114494 A114495

Keyword

nonn,tabf

Author

Emeric Deutsch, Dec 01 2005

Status

approved