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A028364
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T(n, m) equals Sum Catalan(n-k)*Catalan(k), k=0..m.
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24
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1, 1, 2, 2, 3, 5, 5, 7, 9, 14, 14, 19, 23, 28, 42, 42, 56, 66, 76, 90, 132, 132, 174, 202, 227, 255, 297, 429, 429, 561, 645, 715, 785, 869, 1001, 1430, 1430, 1859, 2123, 2333, 2529, 2739, 3003, 3432, 4862
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 0,3
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COMMENTS
| There are several versions of a Catalan triangle: see A009766, A008315, A028364.
The subtriangle [1], [2, 3], [5, 7, 9], ..., namely T(N,M-1), for N>=1, M=1,..,N, appears as one-point function in the totally asymmetric exclusion process for the parameters alpha=1=beta. See the Derrida et al. and Liggett references given under A067323, where these triangle entries are called T_{N,N+M-1} for the given alpha and beta values. See the row reversed triangle A067323.
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REFERENCES
| A. Sapounakis et al., Ordered trees and the inorder transversal, Disc. Math., 306 (2006), 1732-1741.
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FORMULA
| (n, m)-th entry in triangle is Sum Catalan(n-k)*Catalan(k), k=0..m.
T(n, k) = Sum_{j>=0} A039598(k, j)*A039599(n-k, j). - DELEHAM Philippe (kolotoko(AT)wanadoo.fr), Feb 18 2004
Sum_{k>=0} T(n, k) = A001700(n). T(n, k) = A067323(n, n-k), n>=k>=0, else 0 . - Philippe DELEHAM, May 26 2005
Sum_{k>=0} T(n, k) = A001700(n).
G.f. for column sequences m>=0: (-(c(m,x)-1)/x+c(m,x)*c(x))/x^m with the g.f. c(x) of A000108 (Catalan) and c(m,x):=sum(C(k)*x^k,k=0..m) with C(n):=A000108(n). W. Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de), Mar 24 2006.
G.f. for column sequences m>=0 (without leading zeros): c(x)*sum(C(m,k)*c(x)^k,k=0..m) with the g.f. c(x) of A000108 (Catalan) and C(n,m) is the Catalan triangle A033184(n,m). W. Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de), Mar 24 2006.
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EXAMPLE
| {1}, {1, 2}, {2, 3, 5}, {5, 7, 9, 14}, {14, 19, 23, 28, 42}, etc
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MATHEMATICA
| t[n_, k_] = Sum[CatalanNumber[n-j]*CatalanNumber[j], {j, 0, k}]; Flatten[Table[t[n, k], {n, 0, 8}, {k, 0, n}]] (* From Jean-François Alcover, Jul 22 2011 *)
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CROSSREFS
| Cf. A009766, A039598, A039599, A028377, A028378, A028376.
Sequence in context: A113827 A033189 A008507 * A011971 A060048 A110699
Adjacent sequences: A028361 A028362 A028363 * A028365 A028366 A028367
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KEYWORD
| tabl,nonn
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AUTHOR
| Wouter L. J. MEEUSSEN (wouter.meeussen(AT)pandora.be)
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EXTENSIONS
| Name corrected
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