A108747 Triangle read by rows: T(n,k) is the number of Grand Dyck paths of semilength n and having k returns to the x-axis.

2, 2, 4, 4, 8, 8, 10, 20, 24, 16, 28, 56, 72, 64, 32, 84, 168, 224, 224, 160, 64, 264, 528, 720, 768, 640, 384, 128, 858, 1716, 2376, 2640, 2400, 1728, 896, 256, 2860, 5720, 8008, 9152, 8800, 7040, 4480, 2048, 512, 9724, 19448, 27456, 32032, 32032, 27456, 19712, 11264, 4608, 1024

Offset: 1

Author

Emeric Deutsch, Jun 23 2005

Keywords

• nonn
• tabl

Comments

A Grand Dyck path of semilength n is a path in the half-plane x >= 0, starting at (0,0), ending at (2n,0) and consisting of steps u=(1,1) and d=(1,-1).

Triangle T(n,k), 1 <= k <= n, read by rows, given by [0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...] DELTA [2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...] where DELTA is the operator defined in A084938. - Philippe Deléham, Jun 29 2005

T(n,k) is also equal to the number of grand Dyck paths of semilength n having k symmetric vertices. A symmetric vertex is a vertex in the first half of the path (not including the midpoint) that is a mirror image of a vertex in the second half, when with respect to the reflection along the vertical line through the midpoint of the path. - Sergi Elizalde, Feb 12 2021

Examples

			T(2,2)=4 because we have u(d)u(d), u(d)d(u), d(u)d(u) and d(u)u(d) (return steps to x-axis shown between parentheses).
Triangle begins:
   2;
   2,  4;
   4,  8,  8;
  10, 20, 24, 16;
  28, 56, 72, 64, 32;
		

Links

• Sergi Elizalde, The degree of symmetry of lattice paths, arXiv:2002.12874 [math.CO], 2020.
• Sergi Elizalde, Measuring symmetry in lattice paths and partitions, Sem. Lothar. Combin. 84B.26, 12 pp (2020).

Crossrefs

Cf. A000984 (row sums), A000108.

Programs

Maple

T:= (n,k)-> 2^k*k*binomial(2*n-k,n)/(2*n-k): for n from 1 to 10 do seq(T(n,k),k=1..n) od; # yields sequence in triangular form

Mathematica

nn=10;c=(1-(1-4x)^(1/2))/(2x);f[list_]:=Select[list,#>0&];Map[f,Drop[CoefficientList[Series[1/(1-2y x c),{x,0,nn}],{x,y}],1]]//Flatten  (* Geoffrey Critzer, Jan 30 2012 *)

Formula

T(n,1) = 2*A000108(n-1).

T(n,n) = 2^n.

T(n,k) = k * 2^k * binomial(2*n-k,n)/(2*n-k) (1 <= k <= n).

G.f.: 1/(1-2*t*z*C), where C = (1-sqrt(1-4*z))/(2*z) is the Catalan function.

T(n,k) = 2^k * A106566(n,k). - Philippe Deléham, Jun 29 2005