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A078391
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Triangle read by rows: T(n,k) = Catalan(k)*Catalan(n-k).
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2
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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
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OFFSET
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0,4
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COMMENTS
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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
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REFERENCES
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R. Sedgewick and P. Flajolet, Analysis of Algorithms, Addison Wesley, first edition, page 225.
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LINKS
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FORMULA
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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
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
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EXAMPLE
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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
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MATHEMATICA
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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 *)
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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