A101372 Triangle read by rows: T(n,k) is number of leaves at level k in all noncrossing rooted trees on n+1 nodes.

1, 2, 2, 7, 10, 4, 30, 50, 32, 8, 143, 260, 208, 88, 16, 728, 1400, 1280, 704, 224, 32, 3876, 7752, 7752, 5016, 2128, 544, 64, 21318, 43890, 46816, 33880, 17248, 5984, 1280, 128, 120175, 253000, 283360, 222640, 128800, 54400, 16000, 2944, 256

Offset: 1

Author

Emeric Deutsch, Jan 14 2005

Keywords

• nonn
• tabl

Comments

Row n has n terms. Row sums yield A045721. Column 1 is A006013.

Examples

			Triangle begins:
1;
2,2;
7,10,4;
30,50,32,8;
143,260,208,88,16;
...
		

Links

• P. Flajolet and M. Noy, Analytic combinatorics of non-crossing configurations, Discrete Math., 204, 203-229, 1999.
• M. Noy, Enumeration of noncrossing trees on a circle, Discrete Math., 180, 301-313, 1998.

Crossrefs

Cf. A045721, A006013.

Programs

Maple

T:=(n,k)->2^(k-1)*(3*k-1)*binomial(3*n-2,n-k)/(2*n+k-1): for n from 1 to 10 do seq(T(n,k),k=1..n) od; # yields triangle in triangular form

Mathematica

Flatten[Table[2^(k-1) ((3k-1)/(2n+k-1))Binomial[3n-2,n-k],{n,10},{k,n}]] (* Harvey P. Dale, Feb 10 2015 *)

Formula

T(n, k) = 2^(k-1)*[(3k-1)/(2n+k-1)]binomial(3n-2, n-k) (1<=k<=n).

G.f.: t*z*g^2/(1-2*t*z*g^3), where g = 1 + z*g^3 is the g.f. of the ternary numbers (A001764).