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%I #432 Feb 26 2024 11:00:49
%S 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,
%T 1,7,27,75,165,297,429,429,1,8,35,110,275,572,1001,1430,1430,1,9,44,
%U 154,429,1001,2002,3432,4862,4862,1,10,54,208,637,1638,3640,7072,11934
%N 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_{j=0..k} T(n-1,j).
%C The entries in this triangle (in its many forms) are often called ballot numbers.
%C 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
%C 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
%C 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
%C 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
%C 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
%C The triangle sums, see A180662 for their definitions, link Catalan's triangle, its mirror is A033184, with several sequences, see crossrefs. - _Johannes W. Meijer_, Sep 22 2010
%C 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
%C T(n,k) is also the number of nondecreasing parking functions of length n+1 whose maximum element is k+1. For example T(3,2) = 5 because we have (1,1,1,3), (1,1,2,3), (1,2,2,3), (1,1,3,3), (1,2,3,3). - _Ran Pan_, Nov 16 2015
%C T(n,k) is the number of Dyck paths from (0,0) to (n+2,n+2) which start with n-k+2 east steps and touch the diagonal y=x only on the last north step. - _Felipe Rueda_, Sep 18 2019
%C T(n-1,k) for k < n is number of well-formed strings of n parenthesis pairs with prefix of exactly n-k opening parenthesis; T(n,n) = T(n,n-1). - _Hermann Stamm-Wilbrandt_, May 02 2021
%D William Feller, Introduction to Probability Theory and its Applications, vol. I, ed. 2, chap. 3, sect. 1,2.
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%D D. E. Knuth, TAOCP, Vol. 4, Section 7.2.1.6, Eq. 22, p. 451.
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%D M. Bellon, Query 5467, L'Intermédiaire des Mathématiciens, 4 (1925, 11; H. Ory, 4 (1925), 120. - _N. J. A. Sloane_, Mar 09 2022
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%H Luis Verde-Star <a href="https://cs.uwaterloo.ca/journals/JIS/VOL24/Verde/verde4.html">A Matrix Approach to Generalized Delannoy and Schröder Arrays</a>, J. Int. Seq., Vol. 24 (2021), Article 21.4.1.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/CatalansTriangle.html">Catalan's Triangle</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/NonnegativePartialSum.html">Nonnegative Partial Sum</a>
%H Tad White, <a href="https://arxiv.org/abs/2401.01462">Quota Trees</a>, arXiv:2401.01462 [math.CO], 2024. See p. 17.
%H Martha Yip, <a href="https://arxiv.org/abs/1703.00057">Rook Placements and Jordan Forms of Upper-Triangular Nilpotent Matrices</a>, arXiv:1703.00057 [math.CO], 2017; The Electronic Journal of Combinatorics 25(1) (2018), #P1.68.
%F T(n, m) = binomial(n+m, n)*(n-m+1)/(n+1), 0 <= m <= n.
%F 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.
%F 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
%F 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
%F 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
%F Sum_{k=0..floor(n/2)} T(n-k+p-1, k+p-1) = A001405(n+2*p-2) - C(n+2*p-2, p-2), p >= 1. - _Johannes W. Meijer_, Oct 03 2013
%F 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
%F The n-th row polynomial equals the n-th degree Taylor polynomial of the function (1 - 2*x)/(1 - x)^(n+2) about 0. For example, for n = 4, (1 - 2*x)/(1 - x)^6 = 1 + 4*x + 9*x^2 + 14*x^3 + 14*x^4 + O(x^5). - _Peter Bala_, Feb 18 2018
%F T(n,k) = binomial(n + k + 1, k) - 2*binomial(n + k, k - 1), for 0 <= k <= n. - _David Callan_, Jun 15 2022
%e Triangle begins in row n=0 with 0 <= k <= n:
%e 1;
%e 1, 1;
%e 1, 2, 2;
%e 1, 3, 5, 5;
%e 1, 4, 9, 14, 14;
%e 1, 5, 14, 28, 42, 42;
%e 1, 6, 20, 48, 90, 132, 132;
%e 1, 7, 27, 75, 165, 297, 429, 429;
%e 1, 8, 35, 110, 275, 572, 1001, 1430, 1430;
%e 1, 9, 44, 154, 429, 1001, 2002, 3432, 4862, 4862;
%p A009766 := proc(n,k) binomial(n+k,n)*(n-k+1)/(n+1); end proc:
%p seq(seq(A009766(n,k), k=0..n), n=0..10); # _R. J. Mathar_, Dec 03 2010
%t Flatten[NestList[Append[Accumulate[#], Last[Accumulate[#]]] &, {1}, 9]] (* _Birkas Gyorgy_, May 19 2012 *)
%t T[n_, k_] := T[n, k] = Which[k == 0, 1, k>n, 0, True, T[n-1, k] + T[n, k-1] ]; Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Mar 07 2016 *)
%o (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 */
%o (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
%o (Haskell)
%o a009766 n k = a009766_tabl !! n !! k
%o a009766_row n = a009766_tabl !! n
%o a009766_tabl = iterate (\row -> scanl1 (+) (row ++ [0])) [1]
%o -- _Reinhard Zumkeller_, Jul 12 2012
%o (Sage)
%o @CachedFunction
%o def ballot(p,q):
%o if p == 0 and q == 0: return 1
%o if p < 0 or p > q: return 0
%o S = ballot(p-2, q) + ballot(p, q-2)
%o if q % 2 == 1: S += ballot(p-1, q-1)
%o return S
%o A009766 = lambda n, k: ballot(2*k, 2*n)
%o for n in (0..7): [A009766(n, k) for k in (0..n)] # _Peter Luschny_, Mar 05 2014
%o (Sage) [[binomial(n+k,n)*(n-k+1)/(n+1) for k in (0..n)] for n in (0..10)] # _G. C. Greubel_, Mar 07 2019
%o (GAP) Flat(List([0..10],n->List([0..n],m->Binomial(n+m,n)*(n-m+1)/(n+1)))); # _Muniru A Asiru_, Feb 18 2018
%o (Magma) [[Binomial(n+k,n)*(n-k+1)/(n+1): k in [0..n]]: n in [0..10]]; // _G. C. Greubel_, Mar 07 2019
%Y The following are all versions of (essentially) the same Catalan triangle: A009766, A008315, A028364, A030237, A047072, A053121, A059365, A062103, A099039, A106566, A130020, A140344.
%Y Cf. A062745, A214292.
%Y 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, ...
%Y Row sums as well as the sums of the squares of the shallow diagonals give Catalan numbers (A000108).
%Y Reflected version of A033184.
%Y Diagonals give A000108, A000108, A000245, A002057, A000344, A003517, A000588, A003518, A003519, A001392, ...
%Y 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).
%K nonn,tabl,nice
%O 0,5
%A _Wouter Meeussen_
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