A072248 Triangle T(n,k) (n >= 2, 1 <= k <= n-1) giving number of non-crossing trees with n nodes and height k.

1, 1, 2, 1, 7, 4, 1, 20, 26, 8, 1, 54, 126, 76, 16, 1, 143, 548, 504, 200, 32, 1, 376, 2259, 2900, 1656, 496, 64, 1, 986, 9034, 15506, 11528, 4896, 1184, 128, 1, 2583, 35469, 79354, 73172, 39552, 13536, 2752, 256, 1, 6764, 137644, 394642, 439272, 285992, 123904, 35712, 6272, 512

Offset: 2

Author

N. J. A. Sloane, Jul 06 2002

Keywords

• nonn
• tabl

Comments

For n >= 2, the n-th row has n-1 terms.

Examples

			Triangle T(n,k) begins:
1;
1,   2;
1,   7,    4;
1,  20,   26,     8;
1,  54,  126,    76,    16;
1, 143,  548,   504,   200,   32;
1, 376, 2259,  2900,  1656,  496,   64;
1, 986, 9034, 15506, 11528, 4896, 1184, 128;
		

Links

• E. Deutsch and M. Noy, Statistics on non-crossing trees, Discrete Math., 254 (2002), 75-87.

Crossrefs

Cf. A001764, A072247.

Row sums give A001764.

Programs

Maple

T[0]:=z: for k from 1 to 10 do T[k]:=simplify(z/(1-T[k-1]^2/z)) od:for k from 1 to 10 do t[k]:=series(T[k]-T[k-1],z=0,15) od: for n from 2 to 11 do seq(coeff(t[k],z^n),k=1..n-1) od; # Emeric Deutsch, Dec 30 2004

Formula

Column g.f. are T(k) - T(k-1) (k = 1, 2, ...), where T(0) = z and T(k) = z/(1 - T(k-1)^2/z). - Emeric Deutsch, Dec 30 2004

Extensions

More terms from Emeric Deutsch, Dec 30 2004