This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A033184 #284 Jun 11 2025 17:22:30
%S A033184 1,1,1,2,2,1,5,5,3,1,14,14,9,4,1,42,42,28,14,5,1,132,132,90,48,20,6,1,
%T A033184 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_, 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_, 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_, 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. - _Philippe Deléham_, Jun 14 2004
%C A033184 Number of Dyck paths of semilength and having k-1 peaks at height 2. - _Emeric Deutsch_, Aug 31 2004
%C A033184 Riordan array (c(x),x*c(x)), c(x) the g.f. of A000108. Inverse of Riordan array (1-x,x*(1-x)). - _Paul Barry_, Jun 22 2005
%C A033184 Subtriangle of triangle A106566. - _Philippe Deléham_, 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. - _Abdullahi Umar_, Oct 02 2008
%C A033184 Triangle read by rows, product of A065600 and A007318 considered as infinite lower triangular arrays; A033184 = A065600*A007318. - _Philippe Deléham_, Dec 07 2009
%C A033184 The formula stating "Column k is the k-fold convolution of column 1" is equivalent to repeatedly applying M to [1,0,0,0,...], where M is an upper triangular matrix of all 1's with an additional single subdiagonal of 1's. - _Gary W. Adamson_, Jun 06 2011
%C A033184 4^(n-1) = (n-th row terms) dot (first n terms in A001792), where A001792 = binomial transform of the natural numbers: (1, 3, 8, 20, 48, 112, ...). Example: 4^4 = 256 = (14, 14, 9, 4, 1) dot (1, 3, 8, 20, 48) = (42 + 42 + 28 + 14 + 5 + 1) = 256. - _Gary W. Adamson_, Jun 17 2011
%C A033184 The e.g.f. for the n-th subdiagonal of the triangle has the form exp(x)*P(n,x), where P(n,x) is the e.g.f. for row n of triangle A039599. For example, the third row of A039599 is [5, 9, 5, 1] and so the third subdiagonal sequence of this triangle [5, 14, 28, 48, 75, ...] has the e.g.f. exp(x)*(5 + 9*x + 5*x^2/2! + x^3/3!). - _Peter Bala_, Oct 15 2019
%C A033184 Antidiagonals of convolution matrix of Table 1.3, p. 397, of Hoggatt and Bicknell. - _Tom Copeland_, Dec 25 2019
%C A033184 Also the convolution triangle of A120588(n) = A000108(n-1) for n > 0. - _Peter Luschny_, Oct 07 2022
%H A033184 Reinhard Zumkeller, <a href="/A033184/b033184.txt">Rows n = 1..125 of triangle, flattened</a>
%H A033184 José Agapito, Ângela Mestre, Maria M. Torres, and Pasquale Petrullo, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL18/Agapito/agapito2.html">On One-Parameter Catalan Arrays</a>, Journal of Integer Sequences, Vol. 18 (2015), Article 15.5.1.
%H A033184 M. Aigner, <a href="http://dx.doi.org/10.1016/j.disc.2007.06.012">Enumeration via ballot numbers</a>, Discrete Math., 308 (2008), 2544-2563.
%H A033184 J. L. Arregui, <a href="https://arxiv.org/abs/math/0109108">Tangent and Bernoulli numbers related to Motzkin and Catalan numbers by means of numerical triangles</a>, arXiv:math/0109108 [math.NT], 2001.
%H A033184 Peter Bala, <a href="/A033184/a033184.pdf">Factorisations of some Riordan arrays as infinite products</a>
%H A033184 Jean-Luc Baril and Paul Barry, <a href="https://arxiv.org/abs/2212.12404">Two kinds of partial Motzkin paths with air pockets</a>, arXiv:2212.12404 [math.CO], 2022.
%H A033184 Paul Barry, <a href="http://arxiv.org/abs/1107.5490">Invariant number triangles, eigentriangles and Somos-4 sequences</a>, arXiv preprint arXiv:1107.5490 [math.CO], 2011.
%H A033184 Paul Barry, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL14/Barry1/barry97r2.html">Riordan Arrays, Orthogonal Polynomials as Moments, and Hankel Transforms</a>, J. Int. Seq. 14 (2011) # 11.2.2, example 3.
%H A033184 Paul Barry, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL14/Barry3/barry132.html">On the Central Coefficients of Bell Matrices</a>, J. Int. Seq. 14 (2011) # 11.4.3 example 6.
%H A033184 Paul Barry, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL15/Barry4/bern2.html">Riordan-Bernstein Polynomials, Hankel Transforms and Somos Sequences</a>, Journal of Integer Sequences, Vol. 15 2012, #12.8.2.
%H A033184 Paul Barry, <a href="http://arxiv.org/abs/1311.7161">Comparing two matrices of generalized moments defined by continued fraction expansions</a>, arXiv preprint arXiv:1311.7161 [math.CO], 2013 and <a href="https://cs.uwaterloo.ca/journals/JIS/VOL17/Barry3/barry291.html">J. Int. Seq. 17 (2014) # 14.5.1</a>.
%H A033184 Paul Barry, <a href="http://arxiv.org/abs/1312.0583">Embedding structures associated with Riordan arrays and moment matrices</a>, arXiv preprint arXiv:1312.0583 [math.CO], 2013.
%H A033184 Paul Barry, <a href="https://www.emis.de/journals/JIS/VOL22/Barry3/barry422.html">Generalized Catalan Numbers Associated with a Family of Pascal-like Triangles</a>, J. Int. Seq., Vol. 22 (2019), Article 19.5.8.
%H A033184 Paul Barry, <a href="https://arxiv.org/abs/1912.01124">A Note on Riordan Arrays with Catalan Halves</a>, arXiv:1912.01124 [math.CO], 2019.
%H A033184 Paul Barry, <a href="https://arxiv.org/abs/1912.11845">Chebyshev moments and Riordan involutions</a>, arXiv:1912.11845 [math.CO], 2019.
%H A033184 Paul Barry, <a href="https://arxiv.org/abs/2011.13985">The second production matrix of a Riordan array</a>, arXiv:2011.13985 [math.CO], 2020.
%H A033184 Paul Barry, <a href="https://arxiv.org/abs/2409.09547">A Riordan array family for some integrable lattice models</a>, arXiv:2409.09547 [math.CO], 2024. See p. 9.
%H A033184 Paul Barry, <a href="https://doi.org/10.3390/math13020242">Extensions of Riordan Arrays and Their Applications</a>, Mathematics (2025) Vol. 13, No. 2, 242. See p. 13.
%H A033184 Paul Barry, <a href="https://arxiv.org/abs/2504.09719">Notes on Riordan arrays and lattice paths</a>, arXiv:2504.09719 [math.CO], 2025. See pp. 15, 29.
%H A033184 Paul Barry and A. Hennessy, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL13/Barry5/barry96s.html">Meixner-Type Results for Riordan Arrays and Associated Integer Sequences</a>, J. Int. Seq. 13 (2010) # 10.9.4.
%H A033184 Paul Barry and A. Hennessy, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL15/Barry1/barry202.html">Four-term Recurrences, Orthogonal Polynomials and Riordan Arrays</a>, Journal of Integer Sequences, 2012, article 12.4.2. - From _N. J. A. Sloane_, Sep 21 2012
%H A033184 A. Bernini, M. Bouvel and L. Ferrari, <a href="http://puma.dimai.unifi.it/18_3_4/BerniniBouvelFerrari.pdf">Some statistics on permutations avoiding generalized patterns</a>, PU.M.A. Vol. 18 (2007), No. 3-4, pp. 223-237.
%H A033184 S. Brlek, E. Duchi, E. Pergola and S. Rinaldi, <a href="http://dx.doi.org/10.1016/j.disc.2004.07.019">On the equivalence problem for succession rules</a>, Discr. Math., 298 (2005), 142-154.
%H A033184 Fangfang Cai, Qing-Hu Hou, Yidong Sun and Arthur L.B. Yang, <a href="https://arxiv.org/abs/1808.05736">Combinatorial identities related to 2X2 submatrices of recursive matrices</a>, arXiv:1808.05736 [math.CO], 2018.
%H A033184 David Callan, <a href="https://arxiv.org/abs/math/0211380">A recursive bijective approach to counting permutations...</a>, arXiv:math/0211380 [math.CO], 2002.
%H A033184 N. T. Cameron, <a href="https://www.math.hmc.edu/~cameron/dissertation.pdf">Random walks, trees and extensions of Riordan group techniques</a>, Dissertation, Howard University, 2002.
%H A033184 Naiomi Cameron and J. E. McLeod, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL19/McLeod/mcleod3.html">Returns and Hills on Generalized Dyck Paths</a>, Journal of Integer Sequences, Vol. 19, 2016, #16.6.1.
%H A033184 Matteo Cervetti and Luca Ferrari, <a href="https://doi.org/10.1007/s00026-022-00596-1">Enumeration of Some Classes of Pattern Avoiding Matchings, with a Glimpse into the Matching Pattern Poset</a>, Annals of Combinatorics (2022).
%H A033184 Xiaomei Chen, <a href="https://arxiv.org/abs/2412.00668">Counting humps and peaks in Motzkin paths with height k</a>, arXiv:2412.00668 [math.CO], 2024. See p. 6.
%H A033184 Gi-Sang Cheon, Hana Kim and Louis W. Shapiro, <a href="http://dx.doi.org/10.1016/j.disc.2012.03.023">Combinatorics of Riordan arrays with identical A and Z sequences</a>, Discrete Math., 312 (2012), 2040-2049.
%H A033184 Gi-Sang Cheon, S.-T. Jin and L. W. Shapiro, <a href="http://dx.doi.org/10.1016/j.laa.2015.03.015">A combinatorial equivalence relation for formal power series</a>, Linear Algebra and its Applications, Volume 491, 15 February 2016, Pages 123-137; Proceedings of the 19th ILAS Conference, Seoul, South Korea 2014.
%H A033184 Lapo Cioni and Luca Ferrari, <a href="https://arxiv.org/abs/2102.07628">Preimages under the Queuesort algorithm</a>, arXiv preprint arXiv:2102.07628 [math.CO], 2021; Discrete Math., 344 (2021), #112561.
%H A033184 Emeric Deutsch, <a href="http://dx.doi.org/10.1016/S0012-365X(98)00371-9">Dyck path enumeration</a>, Discrete Math., 204, 1999, 167-202.
%H A033184 FindStat - Combinatorial Statistic Finder, <a href="http://www.findstat.org/StatisticsDatabase/St000011">The number of touch points of a Dyck path</a>, <a href="http://www.findstat.org/StatisticsDatabase/St000025">The number of initial rises of a Dyck paths</a>, <a href="http://www.findstat.org/StatisticsDatabase/St000061">The number of nodes on the left branch of the tree</a>, <a href="http://www.findstat.org/StatisticsDatabase/St000084">The number of subtrees</a>.
%H A033184 Shishuo Fu and Yaling Wang, <a href="https://arxiv.org/abs/1908.03912">Bijective recurrences concerning two Schröder triangles</a>, arXiv:1908.03912 [math.CO], 2019.
%H A033184 Samuele Giraudo, <a href="https://arxiv.org/abs/1903.00677">Tree series and pattern avoidance in syntax trees</a>, arXiv:1903.00677 [math.CO], 2019.
%H A033184 Peter M. Higgins, <a href="https://www.researchgate.net/publication/231983398_Combinatorial_results_for_semigroups_of_order-preserving_mappings">Combinatorial results for semigroups of order-preserving mappings</a>, Math. Proc. Camb. Phil. Soc. 113 (1993), 281-296. [From _Abdullahi Umar_, Oct 02 2008]
%H A033184 V. E. Hoggatt, Jr. and M. Bicknell, <a href="https://www.fq.math.ca/Scanned/14-5/hoggatt1.pdf">Catalan and related sequences arising from inverses of Pascal's triangle matrices</a>, Fib. Quart., 14 (1976), 395-405.
%H A033184 JiSun Huh, Sangwook Kim, Seunghyun Seo, and Heesung Shin, <a href="https://arxiv.org/abs/2506.02985">On 102-avoiding inversion sequences</a>, arXiv:2506.02985 [math.CO], 2025. See p. 13.
%H A033184 Wolfdieter 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 Wolfdieter Lang, <a href="https://www.fq.math.ca/Scanned/38-5/lang.pdf">On polynomials related to powers of the generating function of Catalan numbers</a>, The Fibonacci Quart. 38 (2000) 408-19.
%H A033184 P. J. Larcombe and D. R. French, <a href="https://www.researchgate.net/publication/268866232_The_Catalan_number_k-fold_self-convolution_identity_The_original_formulation">The Catalan number k-fold self-convolution identity: the original formulation</a>, Journal of Combinatorial Mathematics and Combinatorial Computing 46 (2003) 191-204.
%H A033184 Toufik Mansour and Mark Shattuck, <a href="https://math.colgate.edu/~integers/z5/z5.pdf">Enumeration of Catalan and smooth words according to capacity</a>, Integers (2025) Vol. 25, Art. No. A5. See p. 12.
%H A033184 R. J. Mathar, <a href="http://arxiv.org/abs/1603.00077">Topologically Distinct Sets of Non-intersecting Circles in the Plane</a>, arXiv:1603.00077 [math.CO], 2016.
%H A033184 D. Merlini, R. Sprugnoli and M. C. Verri, <a href="https://pdfs.semanticscholar.org/d440/77e8b0d03e6a3402e8dfb57b59e6f264064e.pdf">An algebra for proper generating trees</a>
%H A033184 J. Noonan and D. Zeilberger, <a href="https://arxiv.org/abs/math/9808080">The Enumeration of Permutations With a Prescribed Number of ``Forbidden'' Patterns</a>, arXiv:math/9808080 [math.CO], 1998; Also Adv. in Appl. Math. 17 (1996), no. 4, 381--407. MR1422065 (97j:05003).
%H A033184 J.-C. Novelli and J.-Y. Thibon, <a href="https://arxiv.org/abs/math/0512570">Noncommutative Symmetric Functions and Lagrange Inversion</a>, arXiv:math/0512570 [math.CO], 2005-2006.
%H A033184 J. M. Pallo, <a href="https://doi.org/10.1093/comjnl/29.2.171">Enumerating, Ranking and Unranking Binary Trees</a>, The Computer Journal, Volume 29, Issue 2, 1986, Pages 171-175.
%H A033184 R. Pemantle and M. C. Wilson, <a href="https://doi.org/10.1137/050643866">Twenty Combinatorial Examples of Asymptotics Derived from Multivariate Generating Functions</a>, SIAM Rev., 50 (2008), no. 2, 199-272. See p. 264.
%H A033184 A. Reifegerste, <a href="https://arxiv.org/abs/math/0208006">On the diagram of 132-avoiding permutations</a>, arXiv:math/0208006 [math.CO], 2002.
%H A033184 A. Robertson, D. Saracino and D. Zeilberger, <a href="https://arxiv.org/abs/math/0203033">Refined restricted permutations</a>, arXiv:math/0203033 [math.CO], 2002.
%H A033184 Yuriy Shablya, Dmitry Kruchinin and Vladimir Kruchinin, <a href="https://doi.org/10.3390/math8060962">Method for Developing Combinatorial Generation Algorithms Based on AND/OR Trees and Its Application</a>, Mathematics (2020) Vol. 8, No. 6, 962.
%H A033184 Yidong Sun and Fei Ma, <a href="http://arxiv.org/abs/1305.2017">Four transformations on the Catalan triangle</a>, arXiv preprint arXiv:1305.2017 [math.CO], 2013.
%H A033184 Yidong Sun and Fei Ma, <a href="https://doi.org/10.37236/3701">Some new binomial sums related to the Catalan triangle</a>, Electronic Journal of Combinatorics 21(1) (2014), #P1.33.
%H A033184 A. Umar, <a href="http://www.mathnet.ru/adm33">Some combinatorial problems in the theory of symmetric inverse semigroups</a>, Algebra Disc. Math. 9 (2010) 115-126.
%H A033184 Sheng-Liang Yang and L. J. Wang, <a href="https://ajc.maths.uq.edu.au/pdf/64/ajc_v64_p420.pdf">Taylor expansions for the m-Catalan numbers</a>, Australasian Journal of Combinatorics, Volume 64(3) (2016), Pages 420-431.
%H A033184 S.-n. Zheng and S.-l. Yang, <a href="http://dx.doi.org/10.1155/2014/848374">On the-Shifted Central Coefficients of Riordan Matrices</a>, Journal of Applied Mathematics, Volume 2014, Article ID 848374, 8 pages.
%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_, Jul 25 2005
%F A033184 G.f.: t*x*c/(1-t*x*c), where c=(1-sqrt(1-4*x))/(2*x) is the g.f. of the Catalan numbers (A000108). - _Emeric Deutsch_, Mar 01 2004
%F A033184 T(n+1,k+1) = C(2*n-k, n-k)*(k+1)/(n+1). - _Paul D. Hanna_, Aug 11 2008
%F A033184 T((m+1)*n+r-1,m*n+r-1)*r/(m*n+r) = Sum_{k=1..n} (k/n)*T((m+1)*n-k-1,m*n-1)*T(r+k,r), n >= m > 1. - _Vladimir Kruchinin_, Mar 17 2011
%F A033184 T(n-1,m-1) = (m/n)*Sum_{k=1..n-m+1} (k*A000108(k-1)*T(n-k-1,m-2)), n >= m > 1. - _Vladimir Kruchinin_, Mar 17 2011
%F A033184 T(n,k) = C(2*n-k-1,n-k) - C(2*n-k-1,n-k-1). - _Dennis P. Walsh_, Mar 19 2012
%F A033184 T(n,k) = C(2*n-k,n)*k/(2*n-k). - _Dennis P. Walsh_, Mar 19 2012
%F A033184 T(n,k) = T(n,k-1) - T(n-1,k-2). - _Dennis P. Walsh_, Mar 19 2012
%F A033184 G.f.: 2*x*y / (1 + sqrt(1 - 4*x) - 2*x*y) = Sum_{n >= k > 0} T(n, k) * x^n * y^k. - _Michael Somos_, Jun 06 2016
%e A033184 Triangle begins:
%e A033184 ---+-----------------------------------
%e A033184 n\k| 1 2 3 4 5 6 7
%e A033184 ---+-----------------------------------
%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
%e A033184 From _Peter Bala_, Feb 17 2025: (Start)
%e A033184 The array factorizes as an infinite product (read from right to left) of triangular arrays:
%e A033184 / 1 \ / 1 \ / 1 \ / 1 \
%e A033184 | 1 1 | | 0 1 | | 0 1 | | 1 1 |
%e A033184 | 2 2 1 | = ... | 0 0 1 | | 0 1 1 | | 1 1 1 |
%e A033184 | 5 5 3 1 | | 0 0 1 1 | | 0 1 1 1 | | 1 1 1 1 |
%e A033184 |14 14 9 4 1| | 0 0 1 1 1| | 0 1 1 1 1 | | 1 1 1 1 1 |
%e A033184 |... | |... | |... | |... |
%e A033184 See Bala, Example 2.1. (End)
%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);
%p A033184 # Uses function PMatrix from A357368. Adds row and column for n, k = 0.
%p A033184 PMatrix(10, n -> binomial(2*(n-1), n-1) / n); # _Peter Luschny_, Oct 07 2022
%t A033184 nn = 10; c = (1 - (1 - 4 x)^(1/2))/(2 x); f[list_] := Select[list, # > 0 &]; Map[f, Drop[CoefficientList[Series[y x c/(1 - y x c), {x, 0, nn}], {x, y}],1]] //Flatten (* _Geoffrey Critzer_, Jan 31 2012 *)
%t A033184 Flatten[Reverse /@ NestList[Append[Accumulate[#], Last[Accumulate[#]]] &, {1}, 9]] (* _Birkas Gyorgy_, May 19 2012 *)
%t A033184 T[1, 1] := 1; T[n_, k_]/;1<=k<=n := T[n, k] = T[n-1, k-1]+T[n, k+1]; T[n_, k_] := 0; Flatten@Table[T[n, k], {n, 1, 10}, {k, 1, n}] (* _Oliver Seipel_, Dec 31 2024 *)
%o A033184 (PARI) T(n,k)=binomial(2*(n-k)+k,n-k)*(k+1)/(n+1) \\ _Paul D. Hanna_, Aug 11 2008
%o A033184 (Sage) # The simplest way to construct the triangle.
%o A033184 def A033184_triangle(n) :
%o A033184 T = [0 for i in (0..n)]
%o A033184 for k in (1..n) :
%o A033184 T[k] = 1
%o A033184 for i in range(k-1,0,-1) :
%o A033184 T[i] = T[i-1] + T[i+1]
%o A033184 print([T[i] for i in (1..k)])
%o A033184 A033184_triangle(10) # _Peter Luschny_, Jan 27 2012
%o A033184 (Haskell)
%o A033184 a033184 n k = a033184_tabl !! (n-1) !! (k-1)
%o A033184 a033184_row n = a033184_tabl !! (n-1)
%o A033184 a033184_tabl = map reverse a009766_tabl
%o A033184 -- _Reinhard Zumkeller_, Feb 19 2014
%o A033184 (Magma) /* As triangle: */ [[Binomial(2*n-k,n)*k/(2*n-k): k in [1..n]]: n in [1.. 15]]; // _Vincenzo Librandi_, Oct 12 2015
%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), A120588.
%K A033184 nonn,tabl
%O A033184 1,4
%A A033184 _Christian G. Bower_