A118921 Triangle read by rows: T(n,k) is the number of Grand Dyck paths of semilength n having first return to the x-axis at (2k,0) (n,k >= 1). (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)).

2, 4, 2, 12, 4, 4, 40, 12, 8, 10, 140, 40, 24, 20, 28, 504, 140, 80, 60, 56, 84, 1848, 504, 280, 200, 168, 168, 264, 6864, 1848, 1008, 700, 560, 504, 528, 858, 25740, 6864, 3696, 2520, 1960, 1680, 1584, 1716, 2860, 97240, 25740, 13728, 9240, 7056, 5880, 5280

Offset: 1

Author

Emeric Deutsch, May 06 2006

Keywords

• nonn
• tabl

Comments

Row sums are the central binomial coefficients (A000984).

T(n,0) = 2*A028329(n-1).

Sum_{k>=1} k*T(n,k) = 2^(2n-1) (A004171).

For returns to the x-axis arriving from above, see A039599.

Examples

			T(3,2)=4 because we have uudd|ud, uudd|du, dduu|ud and dduu|du (first return to the x-axis shown by | ).
Triangle starts:
    2;
    4,  2;
   12,  4,  4;
   40, 12,  8, 10;
  140, 40, 24, 20, 28;
		

Crossrefs

Cf. A000984, A028329, A004171, A039599.

Programs

Maple

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

Formula

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

G.f. = G(t,z) = (1-sqrt(1-4tz))/sqrt(1-4z).

T(n+1,k+1) = 2*(n-k+1)*A078391(n,k), n >= 0, k >= 0. - Philippe Deléham, Dec 13 2006