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

2,3

Comments

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

Links

Table of n, a(n) for n=2..56.

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

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

Example

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;

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

Crossrefs

Cf. A001764, A072247.

Row sums give A001764.

Sequence in context: A115629 A296461 A144696 * A317360 A177011 A092276

Adjacent sequences: A072245 A072246 A072247 * A072249 A072250 A072251

Keyword

nonn,tabl

Author

N. J. A. Sloane, Jul 06 2002

Extensions

More terms from Emeric Deutsch, Dec 30 2004

Status

approved