0,5

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). - 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))). - 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). - Patrick Labarque, Jul 28 2010

The triangle sums, see A180662 for their definitions, link Catalan's triangle, its mirror is A033184, with several sequences, see the crossrefs. - Johannes W. Meijer, Sep 22 2010

The m-th row of Catalan's triangle consists of the unique nonnegative differences of the form binomial(m+k,m)-binomial(m+k,m+1) with 0<=k<=m (See Links). - R. J. Cano, Jul 22 2014

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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.

J. M. Hammersley, An undergraduate exercise in manipulation, Math. Scientist, 14 (1989), 1-23.

Ki Hang Kim, Douglas G. Rogers, and Fred W. Roush, 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).

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.

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D. Merlini, R. Sprugnoli and M. C. Verri, The tennis ball problem, J. Combin. Theory, A 99 (2002), 307-344 (Table I).

Andrzej Proskurowski and Ekaputra Laiman, 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).

Yidong Sun, A simple bijection between binary trees and colored ternary trees, El. J. Combinat. 17 (2010) #N20

T. D. Noe, Rows n=0..100 of triangle, flattened

Italo J. Dejter, A numeral system for the middle levels, preprint, 2014. [See Section 2. - N. J. A. Sloane, Apr 06 2014]

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Jean-Christophe Aval, Multivariate Fuss-Catalan numbers, Discrete Math., 308 (2008), 4660-4669

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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 Deléham, 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

sum(T(n-k+p-1, k+p-1), k=0..floor(n/2)) = A001405(n+2*p-2) - C(n+2*p-2, p-2), p >= 1. - Johannes W. Meijer, Oct 03 2013

Let A(x,t) denote the o.g.f. Then 1 + x*d/dx(A(x,t))/A(x,t) = 1 + (1 + t)*x + (1 + 2*t + 3*t^2)*x^2 + (1 + 3*t + 6*t^2 + 10*t^3)*x^3 + ... is the o.g.f. for A059481. - Peter Bala, Jul 21 2015

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

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

Flatten[NestList[Append[Accumulate[#], Last[Accumulate[#]]] &, {1}, 9]] (* Birkas Gyorgy, May 19 2012 *)

(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

(Sage)

@CachedFunction

def ballot(p, q):

    if p == 0 and q == 0: return 1

    if p < 0 or p > q: return 0

    S = ballot(p-2, q) + ballot(p, q-2)

    if q % 2 == 1: S += ballot(p-1, q-1)

    return S

A009766 = lambda n, k: ballot(2*k, 2*n)

for n in (0..7): [A009766(n, k) for k in (0..n)]  # Peter Luschny, Mar 05 2014

(PARI) b009766=(n1=0, n2=100)->{my(q=if(!n1, 0, binomial(n1+1, 2))); for(w=n1, n2, for(k=0, w, write("b009766.txt", 1*q" "1*(binomial(w+k, w)-binomial(w+k, w+1))); q++))} \\ R. J. Cano, Jul 22 2014

The following are all versions of (essentially) the same Catalan triangle: A009766, A008315, A028364, A030237, A047072, A053121, A059365, A062103, A099039, A106566, A130020, A140344.

Cf. A062745, A214292.

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, ...

Triangle sums (see the comments): A000108 (Row1), A000957 (Row2), A001405 (Kn11), A014495 (Kn12), A194124 (Kn13), A030238 (Kn21), A000984 (Kn4), A000958 (Fi2), A165407 (Ca1), A026726 (Ca4), A081696 (Ze2).

Sequence in context: A188181 A064581 A064580 * A059718 A076038 A095788

Adjacent sequences:  A009763 A009764 A009765 * A009767 A009768 A009769

nonn,tabl,nice

Wouter Meeussen

approved