A045721 -id:A045721 - cp's OEIS frontend

A100400 Triangle read by rows: T(n,k) is the number of nonroot nodes of outdegree k (0<=k<=n-1) in all non-crossing trees with n edges.

1, 4, 2, 21, 12, 3, 120, 72, 24, 4, 715, 440, 165, 40, 5, 4368, 2730, 1092, 312, 60, 6, 27132, 17136, 7140, 2240, 525, 84, 7, 170544, 108528, 46512, 15504, 4080, 816, 112, 8, 1081575, 692208, 302841, 105336, 29925, 6840, 1197, 144, 9, 6906900, 4440150, 1973400, 708400, 212520, 53130, 10780, 1680, 180, 10

Offset: 1

Emeric Deutsch, Dec 30 2004

Row n contains n terms. Row sums yield A004319. Column 0 yields A045721.

			T(2,1)=2 because in the non-crossing trees /_, _\ and /\ we have 2 nonroot nodes of outdegree 1.
Triangle begins:
1;
4,2;
21,12,3;
120,72,24,4;

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

Cf. A001764, A004319, A045721.

for n from 1 to 10 do seq((k+1)*binomial(3*n-k-2,2*n-1),k=0..n-1) od; # yields sequence in triangular form

T(n, k) = (k+1)binomial(3n-k-2, 2n-1) (0<=k<=n-1).

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

Original entry on oeis.org

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

Emeric Deutsch, Jan 14 2005

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

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

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.

Cf. A045721, A006013.

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

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 *)

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).

cp's OEIS Frontend

A100400 Triangle read by rows: T(n,k) is the number of nonroot nodes of outdegree k (0<=k<=n-1) in all non-crossing trees with n edges.

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