A101401 Triangle read by rows: T(n,k) is the number of noncrossing trees with n edges in which the leftmost child of the root has degree k.

1, 1, 2, 3, 6, 3, 12, 24, 15, 4, 55, 110, 75, 28, 5, 273, 546, 390, 168, 45, 6, 1428, 2856, 2100, 980, 315, 66, 7, 7752, 15504, 11628, 5712, 2040, 528, 91, 8, 43263, 86526, 65835, 33516, 12825, 3762, 819, 120, 9, 246675, 493350, 379500, 198352, 79695, 25410, 6370, 1200, 153, 10

Offset: 1

Author

Emeric Deutsch, Jan 15 2005

Keywords

• nonn
• tabl

Comments

Row n contains n terms. Column k=0 and row sums yield the ternary numbers (A001764).

Examples

			T(2,0)=1 and T(2,1)=2 because the noncrossing trees with 2 edges are /\, |_ and _|.
Triangle starts:
    1;
    1,   2;
    3,   6,   3;
   12,  24,  15,  4;
   55, 110,  75,  28,  5;
  273, 546, 390, 168, 45, 6;
  ...
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 1..1275

Crossrefs

Cf. A001764.

Column k=1 is A046646.

Programs

Maple

T:=proc(n,k) if n=1 and k=1 then 0 elif k<=n then (k+1)*(2*k+1)*binomial(3*n-k-2,2*n-1)/(3*n-k-2) else 0 fi end: for n from 1 to 10 do seq(T(n,k),k=0..n-1) od; # yields sequence in triangular form

Mathematica

T[n_, k_] := ((k + 1)(2k + 1)/(3n - k - 2)) Binomial[3n - k - 2, 2n - 1];
Table[T[n, k], {n, 1, 10}, {k, 0, n - 1}] // Flatten (* Jean-François Alcover, Apr 10 2020 *)

PARI

T(n, k) = if(kAndrew Howroyd, Nov 06 2017

Formula

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

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