A078391 Triangle read by rows: T(n,k) = Catalan(k)*Catalan(n-k).

1, 1, 1, 2, 1, 2, 5, 2, 2, 5, 14, 5, 4, 5, 14, 42, 14, 10, 10, 14, 42, 132, 42, 28, 25, 28, 42, 132, 429, 132, 84, 70, 70, 84, 132, 429, 1430, 429, 264, 210, 196, 210, 264, 429, 1430, 4862, 1430, 858, 660, 588, 588, 660, 858, 1430, 4862, 16796, 4862, 2860, 2145, 1848

Offset: 0

Author

Henry Bottomley, Dec 24 2002

Keywords

• nonn
• tabl

Comments

T(n,k) is the number of Dyck paths of semilength n+1 whose first return point to the axis have abscissa 2k+2. - Emeric Deutsch, Mar 01 2004

With offset = 1, T(n,k) is the number of binary trees with n internal nodes that have exactly k internal nodes in the left subtree, n>=1, 0<=k<=n-1. - Geoffrey Critzer, Feb 24 2013

T(n-1,k) is also the number of tilings of a triangular shape T_n (row k has length k for k=1, 2, ..., n) with n rectangular tiles (including squares) with contain a rectangular tile (n-k,k+1) for k = 0, 1, ... ,n-1, n >= 1. Let the number of tilings of T_n with n rectangular tiles (including squares) be A(n) and take A(0) = 1. Decompose these n-tilings of T_n into n disjoint and exhaustive classes C(n, k), for k = 0, 1, ..., n-1, n >= 1. In class C(n, k) one takes a fixed rectangular tile (n-k,k+1) leaving triangles T_(n-1-k) and T_k to be tiled (but for the k=0 class T_0 is not shown). Then A(n) = A(n-1)*A(0) + A(n-2)*A(1) + ... + A(0)*A(n-1) = sum(A(n-1-k)*A(k), k=0..n-1), n >= 1, with A(0)=1. But this is the recurrence for the Catalan numbers, hence A(n) = C(n). See the link with examples n = 1..7. - Wolfdieter Lang and Kival Ngaokrajang, Dec 27 2014

T(n,k) is the number of triangulations of an (n+3)-polygon using a (0,1,k+2)-triangle. - Yuchun Ji, Jan 21 2021

Alternating sum of even index 2n is A079489(n). - F. Chapoton, Aug 26 2024

Examples

			The triangle T(n,k) begins:
n\k     0    1    2    3    4    5    6    7    8    9    10 ...
0:      1
1:      1    1
2:      2    1    2
3:      5    2    2    5
4:     14    5    4    5   14
5:     42   14   10   10   14   42
6:    132   42   28   25   28   42  132
7:    429  132   84   70   70   84  132  429
8:   1430  429  264  210  196  210  264  429 1430
9:   4862 1430  858  660  588  588  660  858 1430 4862
10: 16796 4862 2860 2145 1848 1764 1848 2145 2860 4862 16796
...  Reformatted - _Wolfdieter Lang_, Dec 27 2014
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References

• R. Sedgewick and P. Flajolet, Analysis of Algorithms, Addison Wesley, first edition, page 225.

Links

• FindStat - Combinatorial Statistic Finder, The position of the first return of a Dyck path., The size of the left subtree.
• Kival Ngaokrajang, Illustration of C(n,k), for n = 1..7, k = 0..n-1
• Richard P. Stanley, Catalan Addendum (k8)

Crossrefs

Row sums are Catalan numbers A000108(n+1). T(2*k,k) = A001246(k), k >= 0. T(n,0) = T(n,n) = A000108(n).

Cf. A067804, A079489.

Programs

Mathematica

nn=10;r=(1-(1-4x)^(1/2))/(2x);l=(1-(1-4x y)^(1/2))/(2x y);f[list_]:=Select[list,#>0&];Map[f,Drop[CoefficientList[Series[1+x l r,{x,0,nn}],{x,y}],1]]//Grid (* Geoffrey Critzer, Feb 24 2013 *)
Table[CatalanNumber[k]CatalanNumber[n-k],{n,0,10},{k,0,n}]//Flatten (* Harvey P. Dale, Nov 14 2019 *)

Formula

G.f.: C(z)*C(tz), where C(z) = (1-sqrt(1-4z))/(2z) is the g.f. of the Catalan numbers (A000108). - Emeric Deutsch, Mar 01 2004

T(n,k) = A118921(n+1,k+1)/(2*(n-k+1)). - Philippe Deléham, Dec 13 2006

When viewed as a square array, for n>0 and k>0, A(n,k) = Sum_{i=0..n-1,j=0..k-1} A[i,j]*A[n-i,k-j]. - Gerald McGarvey, Dec 30 2007