%I A033184
%S A033184 1,1,1,2,2,1,5,5,3,1,14,14,9,4,1,42,42,28,14,5,1,132,132,90,48,20,6,
%T A033184 1,429,429,297,165,75,27,7,1,1430,1430,1001,572,275,110,35,8,1,4862,
%U A033184 4862,3432,2002,1001,429,154,44,9,1
%N A033184 Catalan triangle A009766 transposed.
%C A033184 Triangle read by rows: T(n,k) = number of Dyck n-paths (A000108) containing 
               k returns to ground level. E.g. the paths UDUUDD, UUDDUD each have 
               2 returns; so T(3,2)=2. Row sums over even-indexed columns are the 
               Fine numbers A000957. - David Callan (callan(AT)stat.wisc.edu), Jul 
               25 2005
%C A033184 Triangular array of numbers a(n,k) = number of linear forests of k planted 
               planar trees and n non-root nodes.
%C A033184 Catalan convolution triangle; with offset [0,0]: a(n,m)=(m+1)*binomial(2*n-m,
               n-m)/(n+1), n >= m >= 0, else 0. G.f. for column m: c(x)*(x*c(x))^m 
               with c(x) g.f. for A000108 (Catalan). - Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de), 
               Sep 12 2001
%C A033184 a(n+1,m+1), n >= m >= 0, a(n,m) := 0, n<m, has inverse matrix A030528(n,
               m)*(-1)^(n-m).
%C A033184 a(n,k)=number of Dyck paths of semilength n and having k returns to the 
               axis. Also number of Dyck paths of semilength n and having first 
               peak at height k. Also number of ordered trees with n edges and root 
               degree k. Also number of ordered trees with n edges and having the 
               leftmost leaf at level k. Also number of parallelogram polyominoes 
               of semiperimeter n+1 and having k cells in the leftmost column. - 
               Emeric Deutsch (deutsch(AT)duke.poly.edu), Mar 01 2004
%C A033184 Triangle T(n,k) with 1<=k<=n given by [0, 1, 1, 1, 1, 1, 1, 1, ...] DELTA 
               [1, 0, 0, 0, 0, 0, 0, 0, ...] = 1; 0, 1; 0, 1, 1; 0, 2, 2, 1; 0, 
               5, 5, 3, 1; 0, 14, 14, 9, 4, 1; ... where DELTA is the operator defined 
               in A084938; essentially the same triangle as A059365 . - DELEHAM 
               Philippe (kolotoko(AT)wanadoo.fr), Jun 14 2004
%C A033184 Number of Dyck paths of semilength and having k-1 peaks at height 2. 
               - Emeric Deutsch (deutsch(AT)duke.poly.edu), Aug 31 2004
%C A033184 Riordan array (c(x),xc(x)), c(x) the g.f. of A000108. Inverse of Riordan 
               array (1-x,x(1-x)). - Paul Barry (pbarry(AT)wit.ie), Jun 22 2005
%C A033184 Subtriangle of triangle A106566 . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), 
               Jan 07 2007
%C A033184 T(n, k) is also the number of order-preserving and order-decreasing full 
               transformations (of an n-chain) with exactly k fixed points. [From 
               A. Umar (aumarh(AT)squ.edu.om), Oct 02 2008]
%D A033184 S. Brlek, E. Duchi, E. Pergola and S. Rinaldi, On the equivalence problem 
               for succession rules, Discr. Math., 298 (2005), 142-154.
%D A033184 E. Deutsch, Dyck path enumeration, Discrete Math., 204, 1999, 167-202.
%D A033184 W. Lang, On polynomials related to powers of the generating function 
               of Catalan numbers, The Fibonacci Quart. 38 (2000) 408-19.
%D A033184 P. J. Larcombe and D. R. French, The Catalan number k-fold self-convolution 
               identity: the original formulation, Journal of Combinatorial Mathematics 
               and Combinatorial Computing 46 (2003) 191-204.
%D A033184 M. Aigner, Enumeration via ballot numbers, Discrete Math., 308 (2008), 
               2544-2563.
%D A033184 Higgins, Peter M. Combinatorial results for semigroups of order-preserving 
               mappings. Math. Proc. Camb. Phil. Soc. 113 (1993), 281-296. [From 
               A. Umar (aumarh(AT)squ.edu.om), Oct 02 2008]
%H A033184 J. L. Arregui, <a href="http://arXiv.org/abs/math.NT/0109108">Tangent 
               and Bernoulli numbers</a> related to Motzkin and Catalan numbers 
               by means of numerical triangles.
%H A033184 D. Callan, <a href="http://arXiv.org/abs/math.CO/0211380">A recursive 
               bijective approach to counting permutations...</a>
%H A033184 N. T. Cameron, <a href="http://www.princeton.edu/~wmassey/NAM03/cameron.pdf">
               Random walks, trees and extensions of Riordan group techniques</a>
%H A033184 W. Lang, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">
               On generalizations of Stirling number triangles</a>, J. Integer Seqs., 
               Vol. 3 (2000), #00.2.4.
%H A033184 D. Merlini, R. Sprugnoli and M. C. Verri, <a href="http://www.dsi.unifi.it/
               ~merlini/Lucidi.ps">An algebra for proper generating trees</a>
%H A033184 J. Noonan and D. Zeilberger, <a href="http://arXiv.org/abs/math.CO/9808080">
               [math/9808080] The Enumeration of Permutations With a Prescribed 
               Number of ``Forbidden'' Patterns</a>
%H A033184 A. Reifegerste, <a href="http://arXiv.org/abs/math.CO/0208006">On the 
               diagram of 132-avoiding permutation</a>.
%H A033184 A. Robertson, D. Saracino and D. Zeilberger, <a href="http://arXiv.org/
               abs/math.CO/0203033">Refined restricted permutations</a>.
%H A033184 J.-C. Novelli and J.-Y. Thibon, <a href="http://arXiv.org/abs/math.CO/
               0512570">Noncommutative Symmetric Functions and Lagrange Inversion</
               a>
%F A033184 Column k is the k-fold convolution of column 1. The triangle is also 
               defined recursively by (i) entries outside triangle are 0, (ii) top 
               left entry is 1, (iii) every other entry is sum of its east and northwest 
               neighbor. - David Callan (callan(AT)stat.wisc.edu), Jul 25 2005
%F A033184 G.f.= txc/(1-txc), where c=(1-sqrt(1-4x))/(2x) is the g.f. of the Catalan 
               numbers (A000108). - Emeric Deutsch (deutsch(AT)duke.poly.edu), Mar 
               01 2004
%F A033184 T(n,k) = C(2n-k, n-k)*(k+1)/(n+1). [From Paul D. Hanna (pauldhanna(AT)juno.com), 
               Aug 11 2008]
%e A033184 Triangle begins
%e A033184 \ k..1....2....3....4....5....6
%e A033184 n\
%e A033184 1 |..1
%e A033184 2 |..1....1
%e A033184 3 |..2....2....1
%e A033184 4 |..5....5....3....1
%e A033184 5 |.14...14....9....4....1
%e A033184 6 |.42...42...28...14....5....1
%e A033184 7 |132..132...90...48...20....6....1
%p A033184 a := proc(n,k) if k<=n then k*binomial(2*n-k,n)/(2*n-k) else 0 fi end: 
               seq(seq(a(n,k),k=1..n),n=1..10);
%o A033184 (PARI) T(n,k)=binomial(2*(n-k)+k,n-k)*(k+1)/(n+1) [From Paul D. Hanna 
               (pauldhanna(AT)juno.com), Aug 11 2008]
%Y A033184 Rows of Catalan triangle A009766 read backwards.
%Y A033184 a(n, 1)= A000108(n-1). Row sums = A000108(n) (Catalan).
%Y A033184 The following are all versions of (essentially) the same Catalan triangle: 
               A009766, A030237, A033184, A059365, A099039, A106566, A130020, A047072.
%Y A033184 Diagonals give A000108 A000245 A002057 A000344 A003517 A000588 A003518 
               A003519 A001392, ...
%Y A033184 Cf. A116364 (row squared sums). [From Paul D. Hanna (pauldhanna(AT)juno.com), 
               Aug 11 2008]
%Y A033184 Sequence in context: A114292 A141751 A079222 this_sequence A110488 A134379 
               A108087
%Y A033184 Adjacent sequences: A033181 A033182 A033183 this_sequence A033185 A033186 
               A033187
%K A033184 nonn,tabl
%O A033184 1,4
%A A033184 Christian G. Bower (bowerc(AT)usa.net)