|
| |
|
|
A009766
|
|
Catalan's triangle T(n,k) (read by rows): each term is the sum of the entries above and to the left, i.e. T(n,k)=sum(T(n-1,j),j=0..k).
|
|
83
|
|
|
|
1, 1, 1, 1, 2, 2, 1, 3, 5, 5, 1, 4, 9, 14, 14, 1, 5, 14, 28, 42, 42, 1, 6, 20, 48, 90, 132, 132, 1, 7, 27, 75, 165, 297, 429, 429, 1, 8, 35, 110, 275, 572, 1001, 1430, 1430, 1, 9, 44, 154, 429, 1001, 2002, 3432, 4862, 4862, 1, 10, 54, 208, 637, 1638, 3640, 7072, 11934
(list;
table;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
0,5
|
|
|
COMMENTS
|
The following are all versions of (essentially) the same Catalan triangle: A009766 and A008315, A028364, A030237, A033184, A047072, A053121, A059365, A062103, A099039, A106566, A130020, A140344.
The entries in this triangle (in its many forms) are often called ballot numbers.
T(n,k) = number of standard tableaux of shape (n,k) (n>0, 0< = k< = n). Example: T(3,1) = 3 because we have 134/2, 124/3 and 123/4. - Emeric Deutsch, May 18 2004
T(n,k) is the number of full binary trees with n+1 internal nodes and jump-length k. In the preorder traversal of a full binary tree, any transition from a node at a deeper level to a node on a strictly higher level is called a jump; the positive difference of the levels is called the jump distance; the sum of the jump distances in a given ordered tree is called the jump-length. - Emeric Deutsch, Jan 18 2007
The k-th diagonal from the right (k=1, 2, ...) gives the sequence obtained by asking in how many ways can we toss a fair coin until we first get k more heads than tails. The k-th diagonal has formula k(2m+k-1)!/(m!(m+k)!) and g.f. (C(x))^k where C(x) is the generating function for the Catalan numbers, (1-sqrt(1-4*x))/(2*x). [from Anthony C Robin, Jul 12 2007]
T(n,k) is also the number of order-decreasing and order-preserving full transformations (of an n-element chain) of waist k (waist (alpha) = max(Im(alpha))). [From Abdullahi Umar, Aug 25 2008]
Formatted as an upper right triangle (see tables) a(c,r) is the number of different triangulated planar polygons with c+2 vertices, with triangle degree c-r+1 for the same vertex X (c=column number, r=row number, with c>=r>=1). [From Patrick Labarque (patrick.labarque(AT)base.be), Jul 28 2010]
The triangle sums, see A180662 for their definitions, link Catalan's triangle, its mirror is A033184, with (until now just) nine sequences, see the crossrefs. [Johannes W. Meijer, Sep 22 2010]
|
|
|
REFERENCES
|
Barcucci, Elena; Del Lungo, Alberto; Pergola, Elisa; Pinzani, Renzo. A methodology for plane tree enumeration. Proceedings of the 7th Conference on Formal Power Series and Algebraic Combinatorics (Noisy-le-Grand, 1995). Discrete Math. 180 (1998), no. 1-3, 45--64. MR1603693 (98m:05090) - From N. J. A. Sloane, Jun 03 2012
Eplett, W. J. R. A note about the Catalan triangle. Discrete Math. 25(1979), no. 3, 289--291. MR0534947 (80i:05007)
William Feller, "Introduction to Probability Theory and its Applications", vol. I, ed. 2, chap.3, sect.1,2.
H. G. Forder, Some problems in combinatorics, Math. Gazette, vol. 45, 1961, 199-201.
Aoife Hennessy, A Study of Riordan Arrays with Applications to Continued Fractions, Orthogonal Polynomials and Lattice Paths, Ph. D. Thesis, Waterford Institute of Technology, Oct. 2011; http://repository.wit.ie/1693/1/AoifeThesis.pdf
Kim, Ki Hang; Rogers, Douglas G.; Roush, Fred W. Similarity relations and semiorders. Proceedings of the Tenth Southeastern Conference on Combinatorics, Graph Theory and Computing (Florida Atlantic Univ., Boca Raton, Fla., 1979), pp. 577--594, Congress. Numer., XXIII-XXIV, Utilitas Math., Winnipeg, Man., 1979. MR0561081 (81i:05013) - From N. J. A. Sloane, Jun 05 2012
D. E. Knuth, TAOCP, Vol. 4, Section 7.2.1.6, Eq. 22, p. 451.
W. Krandick, Trees and jumps and real roots, J. Computational and Applied Math., 162, 2004, 51-55.
G. Kreweras, Sur les e'ventails de segments, {\em Cahiers du Bureau Universitaire de Recherche Op\'{e}rationnelle}, Institut de Statistique, Universit\'{e} de Paris, #15 (1970), 3-41.
Laradji, A. and Umar, A. On certain finite semigroups of order-decreasing transformations I, Semigroup Forum 69 (2004), 184-200 [From Abdullahi Umar, Aug 25 2008]
D. Merlini et al., Underdiagonal lattice paths with unrestricted steps, Discrete Appl. Math., 91 (1999), 197-213.
D. Merlini, R. Sprugnoli and M. C. Verri, The tennis ball problem, J. Combin. Theory, A 99 (2002), 307-344 (Table I).
Proskurowski, Andrzej; Laiman, Ekaputra. Fast enumeration, ranking, and unranking of binary trees. Proceedings of the thirteenth Southeastern conference on combinatorics, graph theory and computing (Boca Raton, Fla., 1982). Congr. Numer. 35 (1982), 401--413.MR0725898 (85a:68152). - _N, J. A. Sloane_, Apr 08 2012
Yidong Sun, A simple bijection between binary trees and colored ternary trees, El. J. Combinat. 17 (2010) #N20
|
|
|
LINKS
|
T. D. Noe, Rows n=0..100 of triangle, flattened
J. L. Arregui, Tangent and Bernoulli numbers related to Motzkin and Catalan numbers by means of numerical triangles.
Jean-Christophe Aval, Multivariate Fuss-Catalan numbers, arXiv:0711.0906.
Jean-Christophe Aval, Multivariate Fuss-Catalan numbers, Discrete Math., 308 (2008), 4660-4669
E. Barcucci and Verri, Some more properties of Catalan numbers, Discrete Math., 102 (1992), 229-237.
F. R. Bernhart, Catalan, Motzkin and Riordan numbers, Discr. Math., 204 (1999) 73-112.
M. Bousquet-M\'{e}lou and M. Petkovsek, Linear recurrences with constant coefficients: the multivariate case, Discrete Math. 225 (2000), 51-75.
E. H. M. Brietzke, An identity of Andrews and a new method for the Riordan array proof of combinatorial identities, Discrete Math., 308 (2008), 4246-4262.
S. Brlek, E. Duchi, E. Pergola and S. Rinaldi, On the equivalence problem for succession rules, Discr. Math., 298 (2005), 142-154.
L. Carlitz, Sequences, paths, ballot numbers
B. Derrida, E. Domany and D. Mukamel, An exact solution of a one-dimensional asymmetric exclusion model with open boundaries, J. Stat. Phys. 69, 1992, 667-687; eqs. (20), (21), p. 672. (Y_{N}(K) = A009766(N+1,K-1), 1 <= K <= N+1, N >=0 if alpha = 1 = beta).
E. Deutsch and L. Shapiro, A survey of the Fine numbers, Discrete Math., 241 (2001), 241-265.
D. Foata, G-N. Han, The doubloon polynomial triangle, Ram. J. 23 (2010), 107-126
Dominique Foata and Guo-Niu Han, Doubloons and new q-tangent numbers, Quart. J. Math. 62 (2) (2011) 417-432
Ira Gessel, Super ballot numbers.
F. Hivert, J.-C. Novelli and J.-Y. Thibon, The Algebra of Binary Search Trees, Theoretical Computer Science, 339 (2005), 129-165.
A. Karttunen, Some notes on Catalan's Triangle
D. Merlini et al., Underdiagonal lattice paths with unrestricted steps, Discrete Appl. Math., 91 (1999), 197-213.
J.-C. Novelli and J.-Y. Thibon, Noncommutative Symmetric Functions and Lagrange Inversion, arXiv:math.CO/0512570
R. Parviainen, Permutations, cycles and the pattern 2-13
Planet Math, Lattice Paths
A. Robertson, D. Saracino and D. Zeilberger, Refined restricted permutations.
L. W. Shapiro, A Catalan triangle, Discrete Math., 14, 83-90, 1976.
T. Sillke, Catalan's numbers
R. A. Sulanke, Guessing, ballot numbers and refining Pascal's triangle
Eric Weisstein's World of Mathematics, Catalan's Triangle
Eric Weisstein's World of Mathematics, Nonnegative Partial Sum
|
|
|
FORMULA
|
a(n, m)= binomial(n+m, n)*(n-m+1)/(n+1), 0 <= m <= n.
G.f. for column m: (x^m)*N(2; m-1, x)/(1-x)^(m+1), m >= 0, with the row polynomials from triangle A062991 and N(2; -1, x) := 1.
G.f. C(t*x)/(1-x*C(t*x)) = 1+(1+t)*x+(1+2*t+2*t^2)*x^2+..., where C(x)=(1-sqrt(1-4*x))/(2*x) is the Catalan function. - Emeric Deutsch, May 18 2004
Another version of triangle T(n, k) given by [1, 0, 0, 0, 0, 0, ...] DELTA [0, 1, 1, 1, 1, 1, 1, ...] = 1; 1, 0; 1, 1, 0; 1, 2, 2, 0; 1, 3, 5, 5, 0; 1, 4, 9, 14, 14, 0; ...where DELTA is the operator defined in A084938. - Philippe Deleham (kolotoko(AT)wanadoo.fr) Feb 16 2005
O.g.f. (with offset 1) is the series reversion of x*(1+x*(1-t))/(1+x)^2 = x - x^2*(1+t) + x^3*(1+2*t) - x^4*(1+3*t) + .... - Peter Bala, Jul 15 2012
|
|
|
EXAMPLE
|
Triangle begins in row n=0 with 0<=k<=n:
1
1,1
1,2,2
1,3,5,5
1,4,9,14,14
1,5,14,28,42,42
1,6,20,48,90,132,132
1,7,27,75,165,297,429,429
1,8,35,110,275,572,1001,1430,1430
1,9,44,154,429,1001,2002,3432,4862,4862
|
|
|
MAPLE
|
A009766 := proc(n, k) binomial(n+k, n)*(n-k+1)/(n+1) ; end proc:
seq(seq( A009766(n, k), k=0..n), n=0..10) ; # R. J. Mathar, Dec 03 2010
|
|
|
MATHEMATICA
|
Flatten[NestList[Append[Accumulate[#], Last[Accumulate[#]]] &, {1}, 9]] (* Gyorgy Birkas, May 19 2012 *)
|
|
|
PROG
|
(PARI) {T(n, k)= if(k<0|k>n, 0, binomial(n+1+k, k)* (n+1-k)/(n+1+k) )} /* Michael Somos Oct 17 2006 */
(Haskell)
a009766 n k = a009766_tabl !! n !! k
a009766_row n = a009766_tabl !! n
a009766_tabl = iterate (\row -> scanl1 (+) (row ++ [0])) [1]
-- Reinhard Zumkeller, Jul 12 2012
|
|
|
CROSSREFS
|
The following are all versions of (essentially) the same Catalan triangle: A009766 and A008315, A028364, A030237, A033184, A047072, A053121, A059365, A062103, A099039, A106566, A130020.
Cf. A062745.
Sums of the shallow diagonals give A001405, central binomial coefficients: 1=1, 1=1, 1+1=2, 1+2=3, 1+3+2=6, 1+4+5=10, 1+5+9+5=20, ...
Row sums as well as the sums of the squares of the shallow diagonals give Catalan numbers (A000108).
Reflected version of A033184.
Diagonals give A000108 A000245 A002057 A000344 A003517 A000588 A003518 A003519 A001392, ...
Contribution from Johannes W. Meijer, Sep 22 2010: (Start)
Triangle sums (see the comments): A000108 (Row1); A001405 (Kn11); A014495 (Kn12); A030238 (Kn21); A000984 (Kn4); A000958 (Fi2); A165407 (Ca1); A026726 (Ca4); A081696 (Ze2).
(End)
Cf. A214292.
Sequence in context: A188181 A064581 A064580 * A059718 A076038 A095788
Adjacent sequences: A009763 A009764 A009765 * A009767 A009768 A009769
|
|
|
KEYWORD
|
nonn,tabl,nice
|
|
|
AUTHOR
|
Wouter Meeussen.
|
|
|
STATUS
|
approved
|
| |
|
|
