A080243 -id:A080243 - cp's OEIS frontend

A001003 Schroeder's second problem (generalized parentheses); also called super-Catalan numbers or little Schroeder numbers.

1, 1, 3, 11, 45, 197, 903, 4279, 20793, 103049, 518859, 2646723, 13648869, 71039373, 372693519, 1968801519, 10463578353, 55909013009, 300159426963, 1618362158587, 8759309660445, 47574827600981, 259215937709463, 1416461675464871

Offset: 0

N. J. A. Sloane

If you are looking for the Schroeder numbers (a.k.a. large Schroder numbers, or big Schroeder numbers), see A006318.
Yang & Jiang (2021) call these the small 2-Schroeder numbers. - N. J. A. Sloane, Mar 28 2021
There are two schools of thought about the index for the first term. I prefer the indexing a(0) = a(1) = 1, a(2) = 3, a(3) = 11, etc.
a(n) is the number of ways to insert parentheses in a string of n+1 symbols. The parentheses must be balanced but there is no restriction on the number of pairs of parentheses. The number of letters inside a pair of parentheses must be at least 2. Parentheses enclosing the whole string are ignored.
Also length of list produced by a variant of the Catalan producing iteration: replace each integer k with the list 0,1,..,k,k+1,k,...,1,0 and get the length a(n) of the resulting (flattened) list after n iterations. - Wouter Meeussen, Nov 11 2001
Stanley gives several other interpretations for these numbers.
Number of Schroeder paths of semilength n (i.e., lattice paths from (0,0) to (2n,0), with steps H=(2,0), U=(1,1) and D(1,-1) and not going below the x-axis) with no peaks at level 1. Example: a(2)=3 because among the six Schroeder paths of semilength two HH, UHD, UUDD, HUD, UDH and UDUD, only the first three have no peaks at level 1. - Emeric Deutsch, Dec 27 2003
a(n+1) is the number of Dyck n-paths in which the interior vertices of the ascents are colored white or black. - David Callan, Mar 14 2004
Number of possible schedules for n time slots in the first-come first-served (FCFS) printer policy.
Also row sums of A086810, A033282, A126216. - Philippe Deléham, May 09 2004
a(n+1) is the number of pairs (u,v) of same-length compositions of n, 0's allowed in u but not in v and u dominates v (meaning u_1 >= v_1, u_1 + u_2 >= v_1 + v_2 and so on). For example, with n=2, a(3) counts (2,2), (1+1,1+1), (2+0,1+1). - David Callan, Jul 20 2005
The big Schroeder number (A006318) is the number of Schroeder paths from (0,0) to (n,n) (subdiagonal paths with steps (1,0) (0,1) and (1,1)). These paths fall in two classes: those with steps on the main diagonal and those without. These two classes are equinumerous and the number of paths in either class is the little Schroeder number a(n) (half the big Schroeder number). - Marcelo Aguiar (maguiar(AT)math.tamu.edu), Oct 14 2005
With offset 0, a(n) = number of (colored) Motzkin (n-1)-paths with each upstep U getting one of 2 colors and each flatstep F getting one of 3 colors. Example. With their colors immediately following upsteps/flatsteps, a(2) = 3 counts F1, F2, F3 and a(3)=11 counts U1D, U2D, F1F1, F1F2, F1F3, F2F1, F2F2, F2F3, F3F1, F3F2, F3F3. - David Callan, Aug 16 2006
Shifts left when INVERT transform applied twice. - Alois P. Heinz, Apr 01 2009
Number of increasing tableaux of shape (n,n). An increasing tableau is a semistandard tableaux with strictly increasing rows and columns, and set of entries an initial segment of the positive integers. Example: a(2) = 3 because of the three tableaux (12)(34), (13)(24), (12)(23). - Oliver Pechenik, Apr 22 2014
Number of ordered trees with no vertex of outdegree 1 and having n+1 leaves (called sometimes Schröder trees). - Emeric Deutsch, Dec 13 2014
Number of dissections of a convex (n+2)-gon by nonintersecting diagonals. Example: a(2)=3 because for a square ABCD we have (i) no diagonal, (ii) dissection with diagonal AC, and (iii) dissection with diagonal BD. - Emeric Deutsch, Dec 13 2014
The little Schroeder numbers are the moments of the Marchenko-Pastur law for the case c=2 (although the moment m0 is 1/2 instead of 1): 1/2, 1, 3, 11, 45, 197, 903, ...  - Jose-Javier Martinez, Apr 07 2015
Number of generalized Motzkin paths with no level steps at height 0, from (0,0) to (2n,0), and consisting of steps U=(1,1), D=(1,-1) and H2=(2,0). For example, for n=3, we have the 11 paths: UDUDUD, UUDDUD, UDUUDD, UH2DUD, UDUH2D, UH2H2D, UUDUDD, UUUDDD, UUH2DD, UUDH2D, UH2UDD. - José Luis Ramírez Ramírez, Apr 20 2015
REVERT transform of A225883. - Vladimir Reshetnikov, Oct 25 2015
Total number of (nonempty) faces of all dimensions in the associahedron K_{n+1} of dimension n-1. For example, K_4 (a pentagon) includes 5 vertices and 5 edges together with K_4 itself (5 + 5 + 1 = 11), while K_5 includes 14 vertices, 21 edges and 9 faces together with K_5 itself (14 + 21 + 9 + 1 = 45). - Noam Zeilberger, Sep 17 2018
a(n) is the number of interval posets of permutations with n minimal elements that have exactly two realizers, up to a shift by 1 in a(4). See M. Bouvel, L. Cioni, B. Izart, Theorem 17 page 13. - Mathilde Bouvel, Oct 21 2021
a(n) is the number of sequences of nonnegative integers (u_1, u_2, ..., u_n) such that (i) u_1 = 1, (ii) u_i <= i for all i, (iii) the nonzero u_i are weakly increasing. For example, a(2) = 3 counts 10, 11, 12, and a(3) = 11 counts 100, 101, 102, 103, 110, 111, 112, 113, 120, 122, 123. See link below. - David Callan, Dec 19 2021
a(n) is the number of parking functions of size n avoiding the patterns 132 and 213. - Lara Pudwell, Apr 10 2023
a(n+1) is the number of Schroeder paths from (0,0) to (2n,0) in which level steps at height 0 come in 2 colors. - Alexander Burstein, Jul 23 2023

			G.f. = 1 + x + 3*x^2 + 11*x^3 + 45*x^4 + 197*x^5 + 903*x^6 + 4279*x^7 + ...
a(2) = 3: abc, a(bc), (ab)c; a(3) = 11: abcd, (ab)cd, a(bc)d, ab(cd), (ab)(cd), a(bcd), a(b(cd)), a((bc)d), (abc)d, (a(bc))d, ((ab)c)d.
Sum over partitions formula: a(3) = 2*a(0)*a(2) + 1*a(1)^2 + 3*(a(0)^2)*a(1) + 1*a(0)^4 = 6 + 1 + 3 + 1 = 11.
a(4) = 45 since the top row of Q^3 = (11, 14, 12, 8, 0, 0, 0, ...); (11 + 14 + 12 + 8) = 45.

D. Arques and A. Giorgetti, Une bijection géometrique entre une famille d'hypercartes et une famille de polygones énumérées par la série de Schroeder, Discrete Math., 217 (2000), 17-32.
Paul Barry, Riordan arrays, generalized Narayana triangles, and series reversion, Linear Algebra and its Applications, 491 (2016) 343-385.
N. Bernasconi et al., On properties of random dissections and triangulations, Combinatorica, 30 (6) (2010), 627-654.
Miklos Bona, editor, Handbook of Enumerative Combinatorics, CRC Press, 2015, page 618.
Peter J. Cameron, Some treelike objects. Quart. J. Math. Oxford Ser. (2) 38 (1987), no. 150, 155--183. MR0891613 (89a:05009). See p. 155, also p. 179, line -9. - N. J. A. Sloane, Apr 18 2014
C. Coker, A family of eigensequences, Discrete Math. 282 (2004), 249-250.
L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 57.
D. E. Davenport, L. W. Shapiro and L. C. Woodson, The Double Riordan Group, The Electronic Journal of Combinatorics, 18(2) (2012), #P33. - From N. J. A. Sloane, May 11 2012
Emeric Deutsch, A bijective proof of an equation linking the Schroeder numbers, large and small, Discrete Math., 241 (2001), 235-240.
Tomislav Doslic and Darko Veljan, Logarithmic behavior of some combinatorial sequences. Discrete Math. 308 (2008), no. 11, 2182--2212. MR2404544 (2009j:05019) - From N. J. A. Sloane, May 01 2012
Michael Drmota, Anna de Mier, and Marc Noy, Extremal statistics on non-crossing configurations. Discrete Math. 327 (2014), 103--117. MR3192420. See Eq. (2). - N. J. A. Sloane, May 18 2014
I. M. H. Etherington, On non-associative combinations, Proc. Royal Soc. Edinburgh, 59 (Part 2, 1938-39), 153-162.
I. M. H. Etherington, Some problems of non-associative combinations (I), Edinburgh Math. Notes, 32 (1940), pp. i-vi. Part II is by A. Erdelyi and I. M. H. Etherington, and is on pages vii-xiv of the same issue.
P. Flajolet and M. Noy, Analytic combinatorics of non-crossing permutations, Discrete Math., 204 (1999), 203-229, Section 3.1.
D. Foata and D. Zeilberger, A classic proof of a recurrence for a very classical sequence, J. Comb Thy A 80 380-384 1997.
Wolfgang Gatterbauer and Dan Suciu, Dissociation and propagation for approximate lifted inference with standard relational database management systems, The VLDB Journal, February 2017, Volume 26, Issue 1, pp. 5-30; DOI 10.1007/s00778-016-0434-5
Ivan Geffner and Marc Noy, Counting Outerplanar Maps, Electronic Journal of Combinatorics 24(2) (2017), #P2.3.
D. Gouyou-Beauchamps and B. Vauquelin, Deux propriétés combinatoires des nombres de Schroeder, Theor. Inform. Appl., 22 (1988), 361-388.
N. S. S. Gu, N. Y. Li and T. Mansour, 2-Binary trees: bijections and related issues, Discr. Math., 308 (2008), 1209-1221.
P.-Y. Huang, S.-C. Liu, and Y.-N. Yeh, Congruences of Finite Summations of the Coefficients in certain Generating Functions, The Electronic Journal of Combinatorics, 21 (2014), #P2.45.
M. Klazar, On numbers of Davenport-Schinzel sequences, Discr. Math., 185 (1998), 77-87.
D. E. Knuth, The Art of Computer Programming, Vol. 1, various sections (e.g. p. 534 of 2nd ed.).
D. E. Knuth, The Art of Computer Programming, Vol. 1, (p. 539 of 3rd ed.).
D. E. Knuth, The Art of Computer Programming, Vol. 4A, Section 7.2.1.6, Problem 66, p. 479.
J. S. Lew, Polynomial enumeration of multidimensional lattices, Math. Systems Theory, 12 (1979), 253-270.
Ana Marco and J.-J. Martinez, A total positivity property of the Marchenko-Pastur Law, Electronic Journal of Linear Algebra, 30 (2015), #7.
T. S. Motzkin, Relations between hypersurface cross ratios and a combinatorial formula for partitions of a polygon, for permanent preponderance and for non-associative products, Bull. Amer. Math. Soc., 54 (1948), 352-360.
L. Ozsvart, Counting ordered graphs that avoid certain subgraphs, Discr. Math., 339 (2016), 1871-1877.
R. C. Read, On general dissections of a polygon, Aequat. Mathem. 18 (1978) 370-388, Table 6
J. Riordan, Combinatorial Identities, Wiley, 1968, p. 168.
E. Schroeder, Vier combinatorische Probleme, Zeit. f. Math. Phys., 15 (1870), 361-376.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see page 178; see page 239, Exercise 6.39b.
H. N. V. Temperley and D. G. Rogers, A note on Baxter's generalization of the Temperley-Lieb operators, pp. 324-328 of Combinatorial Mathematics (Canberra, 1977), Lect. Notes Math. 686, 1978.
I. Vardi, Computational Recreations in Mathematica. Addison-Wesley, Redwood City, CA, 1991, p. 198.
Sheng-Liang Yang and Mei-yang Jiang, The m-Schröder paths and m-Schröder numbers, Disc. Math. (2021) Vol. 344, Issue 2, 112209. doi:10.1016/j.disc.2020.112209. See Table 1.

Seiichi Manyama, Table of n, a(n) for n = 0..1312 (first 200 terms from T. D. Noe)
J. Abate and W. Whitt, Integer Sequences from Queueing Theory, J. Int. Seq. 13 (2010), 10.5.5, p_n(1).
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
Ayomikun Adeniran and Lara Pudwell, Pattern avoidance in parking functions, Enumer. Comb. Appl. 3:3 (2023), Article S2R17.
Marcelo Aguiar and Walter Moreira, Combinatorics of the free Baxter algebra, arXiv:math/0510169 [math.CO], 2005-2007, see Corollary 3.3.iii.
Yu Hin Au, Some Properties and Combinatorial Implications of Weighted Small Schröder Numbers, arXiv:1912.00555 [math.CO], 2019.
Axel Bacher, Directed and multi-directed animals on the king's lattice, arXiv preprint arXiv:1301.1365 [math.CO], 2013. - From _N. J. A. Sloane_, Feb 04 2013
Axel Bacher, Improving the Florentine algorithms: recovering algorithms for Motzkin and Schröder paths, arXiv:1802.06030 [cs.DS], 2018.
C. Banderier, M. Bousquet-Mélou, A. Denise, P. Flajolet, D. Gardy and D. Gouyou-Beauchamps, Generating functions for generating trees, Discrete Mathematics 246(1-3), March 2002, pp. 29-55.
C. Banderier and D. Merlini, Lattice paths with an infinite set of jumps, FPSAC02, Melbourne, 2002.
E. Barcucci, E. Pergola, R. Pinzani and S. Rinaldi, ECO method and hill-free generalized Motzkin paths, Séminaire Lotharingien de Combinatoire, B46b (2001), 14 pp.
Marilena Barnabei, Flavio Bonetti, Niccolò Castronuovo and Matteo Silimbani, Ascending runs in permutations and valued Dyck paths, Ars Mathematica Contemporanea (2019) Vol. 16, No. 2, 445-463.
Paul Barry, On Integer-Sequence-Based Constructions of Generalized Pascal Triangles, J. Integer Sequ., Vol. 9 (2006), Article 06.2.4.
Paul Barry, Generalized Catalan Numbers, Hankel Transforms and Somos-4 Sequences , J. Int. Seq. 13 (2010) #10.7.2.
Paul Barry, Comparing two matrices of generalized moments defined by continued fraction expansions, arXiv preprint arXiv:1311.7161 [math.CO], 2013 and J. Int. Seq. 17 (2014) # 14.5.1.
Paul Barry, On a transformation of Riordan moment sequences, arXiv:1802.03443 [math.CO], 2018.
Paul Barry, Generalized Catalan recurrences, Riordan arrays, elliptic curves, and orthogonal polynomials, arXiv:1910.00875 [math.CO], 2019.
Paul Barry and Aoife Hennessy, Generalized Narayana Polynomials, Riordan Arrays, and Lattice Paths, Journal of Integer Sequences, Vol. 15, 2012, #12.4.8.
Karin Baur and P. P. Martin, The fibres of the Scott map on polygon tilings are the flip equivalence classes, arXiv preprint arXiv:1601.05080 [math.CO], 2016.
Karin Baur and Paul P. Martin, A generalised Euler-Poincaré formula for associahedra, arXiv:1711.04986 [math.CO], 2017.
Arkady Berenstein, Vladimir Retakh, Christophe Reutenauer and Doron Zeilberger, The Reciprocal of Sum_{n >= 0} a^n b^n for non-commuting a and b, Catalan numbers and non-commutative quadratic equations, arXiv preprint arXiv:1206.4225 [math.CO], 2012.
Julia E. Bergner, Cedric Harper, Ryan Keller and Mathilde Rosi-Marshall, Action graphs, planar rooted forests, and self-convolutions of the Catalan numbers, arXiv:1807.03005 [math.CO], 2018.
F. R. Bernhart & N. J. A. Sloane, Emails, April-May 1994
D. Birmajer, J. B. Gil and M. D. Weiner, Colored partitions of a convex polygon by noncrossing diagonals, arXiv preprint arXiv:1503.05242 [math.CO], 2015.
Daniel Birmajer and Juan B. Gil, Michael D. Weiner, A family of Bell transformations, arXiv:1803.07727 [math.CO], 2018.
Natasha Blitvić and Einar Steingrímsson, Permutations, moments, measures, arXiv:2001.00280 [math.CO], 2020.
J. Bloom and A, Burstein, Egge triples and unbalanced Wilf-equivalence, arXiv preprint arXiv:1410.0230 [math.CO], 2014.
Miklós Bóna, Cheyne Homberger, Jay Pantone, and Vince Vatter, Pattern-avoiding involutions: exact and asymptotic enumeration, arxiv:1310.7003 [math.CO], 2013.
Henry Bottomley, Illustration of initial terms
Mathilde Bouvel, Cedric Chauve, Marni Mishna and Dominique Rossin, Average-case analysis of perfect sorting by reversals, arXiv preprint arXiv:1201.0940 [cs.DM], 2012.
M. Bouvel, L. Cioni and B. Izart, The interval posets of permutations seen from the decomposition tree perspective, arXiv:2110.10000 [math.CO], 2021. See Theorem 17 p. 13.
Kevin Brown, Hipparchus on Compound Statements, 1994-2010.
Alexander Burstein and Opel Jones, Enumeration of Dumont permutations avoiding certain four-letter patterns, arXiv:2002.12189 [math.CO], 2020.
Alexander Burstein and Louis W. Shapiro, Pseudo-involutions in the Riordan group, arXiv:2112.11595 [math.CO], 2021.
Freddy Cachazo, Karen Yeats and Samuel Yusim, Compatible Cycles and CHY Integrals, arXiv:1907.12661 [math-ph], 2019.
Fangfang Cai, Qing-Hu Hou, Yidong Sun and Arthur L.B. Yang, Combinatorial identities related to 2X2 submatrices of recursive matrices, arXiv:1808.05736 [math.CO], 2018.
H. Cambazard and N. Catusse, Fixed-Parameter Algorithms for Rectilinear Steiner tree and Rectilinear Traveling Salesman Problem in the Plane, arXiv preprint arXiv:1512.06649 [cs.DS], 2015.
David Callan, Some bijections for lattice paths, arXiv:2112.05241 [math.CO], 2021.
Naiomi T. Cameron and Asamoah Nkwanta, On Some (Pseudo) Involutions in the Riordan Group, Journal of Integer Sequences, Vol. 8 (2005), Article 05.3.7.
P. J. Cameron, Some sequences of integers, Discrete Math., 75 (1989), 89-102.
P. J. Cameron, Some sequences of integers, in "Graph Theory and Combinatorics 1988", ed. B. Bollobas, Annals of Discrete Math., 43 (1989), 89-102.
F. Cazals, Combinatorics of Non-Crossing Configurations, Studies in Automatic Combinatorics, Volume II (1997).
Grégory Chatel, Vincent Pilaud and Viviane Pons, The weak order on integer posets, arXiv:1701.07995 [math.CO], 2017.
W. Y. C. Chen, T. Mansour and S. H. F. Yan, Matchings avoiding partial patterns, arXiv:math/0504342 [math.CO], 2005.
William Y. C. Chen and Carol J. Wang, Noncrossing Linked Partitions and Large (3, 2)-Motzkin Paths, Discrete Math., 312 (2012), 1918-1922. - _N. J. A. Sloane_, Jun 08 2012
Z. Chen and H. Pan, Identities involving weighted Catalan, Schröder and Motzkin Paths, arXiv:1608.02448 [math.CO] (2016), eq (1.13) a=1, b=2.
J. Cigler, Ramanujan's q-continued fractions and Schröder-like numbers, arXiv:1210.0372 [math.HO], 2012. - _N. J. A. Sloane_, Dec 29 2012
J. Cigler, Hankel determinants of some polynomial sequences, arXiv:1211.0816 [math.CO], 2012.
Dennis E. Davenport, Louis W. Shapiro and Leon C. Woodson, A bijection between the triangulations of convex polygons and ordered trees, Integers (2020) Vol. 20, Article #A8.
D. Drake, Bijections from Weighted Dyck Paths to Schröder Paths, J. Int. Seq. 13 (2010) # 10.9.2.
Vesselin Drensky, Graded Algebras, Algebraic Functions, Planar Trees, and Elliptic Integrals, arXiv:2004.05596 [math.RA], 2020, see p. 20.
Rosena R. X. Du, Xiaojie Fan and Yue Zhao, Enumeration on row-increasing tableaux of shape 2 X n, arXiv:1803.01590 [math.CO], 2018.
I. M. H. Etherington, Non-associate powers and a functional equation, Math. Gaz. 21 (1937), 36-39; addendum 21 (1937), 153.
I. M. H. Etherington, Some problems of non-associative combinations (I), Edinburgh Math. Notes, 32 (1940), pp. i-vi. [Annotated scanned copy]. Part II [not scanned] is by A. Erdelyi and I. M. H. Etherington, and is on pages vii-xiv of the same issue.
S.-P. Eu and T.-S. Fu, A simple proof of the Aztec diamond problem, arXiv:math/0412041 [math.CO], 2004.
P. Flajolet and R. Sedgewick, Analytic Combinatorics, 2009; see pages 69, 474-475, etc.
Oisín Flynn-Connolly, E_n-coalgebras in simplicial sets, GitHub, Leiden Univ. (Netherlands, 2025). See p. 24.
D. Foata and D. Zeilberger, A Classic Proof of a Recurrence for a Very Classical Sequence, arXiv:math/9805015 [math.CO], 1998.
D. Foata and D. Zeilberger, A classic proof of a recurrence for a very classical sequence
Shishuo Fu and Yaling Wang, Bijective recurrences concerning two Schröder triangles, arXiv:1908.03912 [math.CO], 2019.
N. Gabriel, K. Peske, L. Pudwell and S. Tay, Pattern Avoidance in Ternary Trees, J. Int. Seq. 15 (2012) # 12.1.5.
W. Gatterbauer and D. Suciu, Approximate Lifted Inference with Probabilistic Databases, arXiv preprint arXiv:1412.1069 [cs.DB], 2014.
Joël Gay and Vincent Pilaud, The weak order on Weyl posets, arXiv:1804.06572 [math.CO], 2018.
Evangelos Georgiadis, Akihiro Munemasa and Hajime Tanaka, A note on super Catalan numbers, arXiv:1101.1579 [math.CO], 2011-2012.
Olivier Gérard, Illustration of initial terms (a)
Olivier Gérard, Illustration of initial terms (b)
Étienne Ghys, A Singular Mathematical Promenade, arXiv:1612.06373, 2016, also, The Mathematical Intelligencer (2018) 40.2, 85-88.
Samuele Giraudo, Pluriassociative algebras II: The polydendriform operad and related operads, arXiv:1603.01394 [math.CO], 2016.
Samuele Giraudo, Tree series and pattern avoidance in syntax trees, arXiv:1903.00677 [math.CO], 2019.
Li Guo and Jun Pei, Averaging algebras, Schröder numbers, rooted trees and operads, arXiv preprint arXiv:1401.7386 [math.RA], 2014.
Guo-Niu Han, Enumeration of Standard Puzzles
Guo-Niu Han, Enumeration of Standard Puzzles [Cached copy]
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.
Cheyne Homberger, Patterns in Permutations and Involutions: A Structural and Enumerative Approach, arXiv preprint 1410.2657 [math.CO], 2014.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 42 [Dead link]
St. John Kettle, Letter to N. J. A. Sloane, 1982.
Anton Khoroshkin and Thomas Willwacher, Real moduli space of stable rational curves revised, arXiv:1905.04499 [math.AT], 2019.
A. Kirillov, On Some Quadratic Algebras I 1/2: Combinatorics of Dunkl and Gaudin Elements, Schubert, Grothendieck, Fuss-Catalan, Universal Tutte and Reduced Polynomials, arXiv preprint arXiv:1502.00426 [math.RT], 2016.
E. Krasko and A. Omelchenko, Brown's Theorem and its Application for Enumeration of Dissections and Planar Trees, The Electronic Journal of Combinatorics, 22 (2015), #P1.17.
G. Kreweras, Sur les hiérarchies de segments, Cahiers du Bureau Universitaire de Recherche Opérationnelle, Institut de Statistique, Université de Paris, #20 (1973).
G. Kreweras, Sur les hiérarchies de segments, Cahiers du Bureau Universitaire de Recherche Opérationnelle, Institut de Statistique, Université de Paris, #20 (1973). (Annotated scanned copy)
G. Kreweras, Les préordres totaux compatibles avec un ordre partiel, Math. Sci. Humaines No. 53 (1976), 5-30.
G. Kreweras, Letter to N. J. A. Sloane, 1978.
V. V. Kruchinin and D. V. Kruchinin, A Method for Obtaining Generating Function for Central Coefficients of Triangles, arXiv preprint arXiv:1206.0877 [math.CO], 2012, and J. Int. Seq. 15 (2012) #12.9.3
V. Kurauskas, On graphs containing few disjoint excluded minors. Asymptotic number and structure of graphs containing few disjoint minors K_4, arXiv preprint arXiv:1504.08107 [math.CO], 2015. See Section 8.1.
J. S. Lew, Letter to N. J. A. Sloane, Sep 05 1978
Huyile Liang, Jeffrey Remmel and Sainan Zheng, Stieltjes moment sequences of polynomials, arXiv:1710.05795 [math.CO], 2017, see page 11.
Peter Luschny, The Lost Catalan Numbers And The Schröder Tableaux.
A. Marco and J.-J. Martinez, A total positivity property of the Marchenko-Pastur Law, Electronic Journal of Linear Algebra, Volume 30 (2015), pp. 106-117.
Peter McCalla and Asamoah Nkwanta, Catalan and Motzkin Integral Representations, arXiv:1901.07092 [math.NT], 2019.
D. Merlini, R. Sprugnoli and M. C. Verri, Waiting patterns for a printer, Discrete Applied Mathematics, Volume 144, Issue 3, 2004, pp. 359-373, ISSN 0166-218X.
W. Mlotkowski and K. A. Penson, The probability measure corresponding to 2-plane trees, ArXiv preprint arXiv:1304.6544 [math.PR], 2013.
T. Motzkin, The hypersurface cross ratio, Bull. Amer. Math. Soc., 51 (1945), 976-984.
T. S. Motzkin, Relations between hypersurface cross ratios and a combinatorial formula for partitions of a polygon, for permanent preponderance and for non-associative products, Bull. Amer. Math. Soc., 54 (1948), 352-360.
Hanna Mularczyk, Lattice Paths and Pattern-Avoiding Uniquely Sorted Permutations, arXiv:1908.04025 [math.CO], 2019.
G. Muntingh, Implicit Divided Differences, Little Schröder Numbers, and Catalan Numbers, arXiv preprint arXiv:1204.2709 [math.CO], 2012, and J. Int. Seq. 15 (2012) #12.6.5.
Jean-Christophe Novelli and Jean-Yves Thibon, Duplicial algebras and Lagrange inversion, arXiv preprint arXiv:1209.5959 [math.CO], 2012.
J.-C. Novelli and J.-Y. Thibon, Hopf Algebras of m-permutations,(m+1)-ary trees, and m-parking functions, arXiv preprint arXiv:1403.5962 [math.CO], 2014.
P. Peart and W.-J. Woan, Generating Functions via Hankel and Stieltjes Matrices, J. Integer Seqs., Vol. 3 (2000), #00.2.1.
O. Pechenik, Cyclic sieving of increasing tableaux and small Schröder paths, arXiv:1209.1355 [math.CO], 2012-2014.
O. Pechenik, Cyclic sieving of increasing tableaux and small Schröder paths, J. Combin. Theory A, 125 (2014), 357-378.
R. A. Perez, A brief but historic article of Siegel, Notices AMS, 58 (2011), 558-566.
E. Pergola and R. A. Sulanke, Schroeder Triangles, Paths, and Parallelogram Polyominoes, J. Integer Sequences, 1 (1998), #98.1.7.
Vincent Pilaud, Pebble trees, arXiv:2205.06686 [math.CO], 2022.
Vincent Pilaud and V. Pons, Permutrees, arXiv preprint arXiv:1606.09643 [math.CO], 2016.
L. Pudwell, Pattern avoidance in trees (slides from a talk, mentions many sequences), 2012.
Feng Qi, Xiao-Ting Shi, and Bai-Ni Guo, Integral representations of the large and little Schröder numbers, Preprint, 2016.
Feng Qi and Bai-Ni Guo, Some explicit and recursive formulas of the large and little Schröder numbers, Arab Journal of Mathematical Sciences, June 2016.
Marko Riedel, Computation of the asymptotic by elementary means
E. Schröder, Vier combinatorische Probleme, Z. f. Math. Phys., 15 (1870), 361-376. [Annotated scanned copy]
L. W. Shapiro & N. J. A. Sloane, Correspondence, 1976
M. Shattuck, On the zeros of some polynomials with combinatorial coefficients, Annales Mathematicae et Informaticae, 42 (2013) pp. 93-101.
N. J. A. Sloane, The On-Line Encyclopedia of Integer Sequences, arXiv:math/0312448 [math.CO], 2003.
N. J. A. Sloane, Transforms
N. J. A. Sloane, Illustrations of rooted planar trees with 2, 3, and 4 endpoints, illustrating a(1)=1, a(2)=3, a(4)=11.
L. M. Smiley, Variants of Schroeder Dissections, arXiv:math/9907057 [math.CO], 1999.
Michael Z. Spivey and Laura L. Steil, The k-Binomial Transforms and the Hankel Transform, Journal of Integer Sequences, Vol. 9 (2006), Article 06.1.1.
R. P. Stanley, Hipparchus, Plutarch, Schröder and Hough, Am. Math. Monthly, Vol. 104, No. 4, p. 344, 1997.
R. P. Stanley, Hipparchus, Plutarch, Schröder, and Hough, Amer. Math. Monthly, 104, 1997, 344-350.
Y. Sun and Z. Wang, Consecutive pattern avoidances in non-crossing trees, Graph. Combinat. 26 (2010) 815-832
Anthony Van Duzer, Subtrees of a Given size in Schroeder Trees, arXiv:1904.05525 [math.CO], 2019.
Elena L. Wang and Guoce Xin, On Ward Numbers and Increasing Schröder Trees, arXiv:2507.15654 [math.CO], 2025. See p. 12.
Eric Weisstein's World of Mathematics, Bracketing
Eric Weisstein's World of Mathematics, Super Catalan Number
Wen-jin Woan, A Recursive Relation for Weighted Motzkin Sequences Journal of Integer Sequences, Vol. 8 (2005), Article 05.1.6.
Wen-jin Woan, A Relation Between Restricted and Unrestricted Weighted Motzkin Paths, Journal of Integer Sequences, Vol. 9 (2006), Article 06.1.7.
Wikipedia, Schroeder-Hipparchus numbers.
E. X. W. Xia and O. X. M. Yao, A Criterion for the Log-Convexity of Combinatorial Sequences, The Electronic Journal of Combinatorics, 20 (2013), #P3.
Index entries for sequences related to parenthesizing
Index entries for "core" sequences

See A000081, A000108, A001190, A001699, for other ways to count parentheses.
Row sums of A033282, A033877, A086810, A126216.
Right-hand column 1 of convolution triangle A011117.
Column 1 of A336573. Column 0 of A104219.
The sequences listed in Yang-Jiang's Table 1 appear to be A006318, this sequence, A027307, A034015, A144097, A243675, A260332, A243676. - N. J. A. Sloane, Mar 28 2021
Cf. A000041, A000311, A001263, A003480, A006125, A010683, A025240, A035011.
Cf. A048996, A054726, A059435, A065096, A080243, A085403, A086456, A086616.
Cf. A086810, A088617, A104858, A110440, A111785, A114710, A118376, A126216.
Cf. A139685, A144156, A144944, A153881, A186826.
Cf. A006318 (Schroeder numbers).

a001003 = last . a144944_row  -- Reinhard Zumkeller, May 11 2013

R:=PowerSeriesRing(Rationals(), 50);
Coefficients(R!( (1+x -Sqrt(1-6*x+x^2) )/(4*x) )); // G. C. Greubel, Oct 27 2024

t1 := (1/(4*x))*(1+x-sqrt(1-6*x+x^2)); series(t1,x,40);
invtr:= proc(p) local b; b:= proc(n) option remember; local i; `if`(n<1, 1, add(b(n-i) *p(i-1), i=1..n+1)) end end: a:='a': f:= (invtr@@2)(a): a:= proc(n) if n<0 then 1 else f(n-1) fi end: seq(a(n), n=0..30); # Alois P. Heinz, Apr 01 2009
# Computes n -> [a[0],a[1],..,a[n]]
A001003_list := proc(n) local j,a,w; a := array(0..n); a[0] := 1;
for w from 1 to n do a[w] := a[w-1]+2*add(a[j]*a[w-j-1],j=1..w-1) od;
convert(a,list) end: A001003_list(100); # Peter Luschny, May 17 2011

Table[Length[Flatten[Nest[ #/.a_Integer:> Join[Range[0, a + 1], Range[a, 0, -1]] &, {0}, n]]], {n, 0, 10}]; Sch[ 0 ] = Sch[ 1 ] = 1; Sch[ n_Integer ] := Sch[ n ] = (3(2n - 1)Sch[ n - 1 ] - (n - 2)Sch[ n - 2 ])/(n + 1); Array[ Sch, 24, 0]
(* Second program: *)
a[n_] := Hypergeometric2F1[-n + 1, n + 2, 2, -1]; a[0] = 1; Table[a[n], {n, 0, 23}] (* Jean-François Alcover, Nov 09 2011, after Vladeta Jovovic *)
a[ n_] := SeriesCoefficient[ (1 + x - Sqrt[1 - 6 x + x^2]) / (4 x), {x, 0, n}]; (* Michael Somos, Aug 26 2015 *)
Table[(KroneckerDelta[n] - GegenbauerC[n+1, -1/2, 3])/4, {n, 0, 20}] (* Vladimir Reshetnikov, Oct 25 2015 *)
a[n_] := -LegendreP[n, -1, 2, 3] I / Sqrt[2]; a[0] = 1;
Table[a[n], {n, 0, 23}] (* Jean-François Alcover, Feb 16 2019 *)
a[1]:=1; a[2]:=1; a[n_]:=a[n] = a[n-1]+2 Sum[a[k] a[n-k], {k,2,n-1}]; Map[a, Range[24]] (* Oliver Seipel, Nov 03 2024, after Schröder 1870 *)
CoefficientList[InverseSeries[Series[x/(Series[(1 - x)/(1 - 2  x), {x, 0, 24}]), {x, 0, 24}]]/x, x] (* Mats Granvik, Jun 30 2025 *)

{a(n) = if( n<1, n==0, sum( k=0, n, 2^k * binomial(n, k) * binomial(n, k-1) ) / (2*n))}; /* Michael Somos, Mar 31 2007 */

{a(n) = my(A); if( n<1, n==0, n--; A = x * O(x^n); n! * simplify( polcoef( exp(3*x + A) * besseli(1, 2*x * quadgen(8) + A), n)))}; /* Michael Somos, Mar 31 2007 */

{a(n) = if( n<0, 0, n++; polcoef( serreverse( (x - 2*x^2) / (1 - x) + x * O(x^n)), n))}; /* Michael Somos, Mar 31 2007 */

N=30; x='x+O('x^N); Vec( (1+x-(1-6*x+x^2)^(1/2))/(4*x) ) \\ Hugo Pfoertner, Nov 19 2018

# The objective of this implementation is efficiency.
# n -> [a(0), a(1), ..., a(n)]
def A001003_list(n):
    a = [0 for i in range(n)]
    a[0] = 1
    for w in range(1, n):
        s = 0
        for j in range(1, w):
            s += a[j]*a[w-j-1]
        a[w] = a[w-1]+2*s
    return a
# Peter Luschny, May 17 2011

from gmpy2 import divexact
A001003 = [1, 1]
for n in range(3,10**3):
    A001003.append(divexact(A001003[-1]*(6*n-9)-(n-3)*A001003[-2],n))
# Chai Wah Wu, Sep 01 2014

# Generalized algorithm of L. Seidel
def A001003_list(n) :
    D = [0]*(n+1); D[1] = 1/2
    b = True; h = 2; R = [1]
    for i in range(2*n-2) :
        if b :
            for k in range(h,0,-1) : D[k] += D[k-1]
            h += 1;
        else :
            for k in range(1,h, 1) : D[k] += D[k-1]
            R.append(D[h-1]);
        b = not b
    return R
A001003_list(24) # Peter Luschny, Jun 02 2012

D-finite with recurrence: (n+1) * a(n) = (6*n-3) * a(n-1) - (n-2) * a(n-2) if n>1. a(0) = a(1) = 1.
a(n) = 3*a(n-1) + 2*A065096(n-2) (n>2). If g(x) = 1 + 3*x + 11*x^2 + 45*x^3 + ... + a(n)*x^n + ..., then g(x) = 1 + 3(x*g(x)) + 2(x*g(x))^2, g(x)^2 = 1 + 6*x + 31*x^2 + 156*x^3 + ... + A065096(n)*x^n + ... - Paul D. Hanna, Jun 10 2002
a(n+1) = -a(n) + 2*Sum_{k=1..n} a(k)*a(n+1-k). - Philippe Deléham, Jan 27 2004
a(n-2) = (1/(n-1))*Sum_{k=0..n-3} binomial(n-1,k+1)*binomial(n-3,k)*2^(n-3-k) for n >= 3 [G. Polya, Elemente de Math., 12 (1957), p. 115.] - N. J. A. Sloane, Jun 13 2015
G.f.: (1 + x - sqrt(1 - 6*x + x^2) )/(4*x) = 2/(1 + x + sqrt(1 - 6*x + x^2)).
a(n) ~ W*(3+sqrt(8))^n*n^(-3/2) where W = (1/4)*sqrt((sqrt(18)-4)/Pi) [See Knuth I, p. 534, or Perez. Note that the formula on line 3, page 475 of Flajolet and Sedgewick seems to be wrong - it has to be multiplied by 2^(1/4).] - N. J. A. Sloane, Apr 10 2011
The correct asymptotic for this sequence is a(n) ~ W*(3+sqrt(8))^n*n^(-3/2), where W = (1+sqrt(2))/(2*2^(3/4)*sqrt(Pi)) = 0.404947065905750651736243... Result in book by D. Knuth (p. 539 of 3rd edition, exercise 12) is for sequence b(n), but a(n) = b(n+1)/2. Therefore is asymptotic a(n) ~ b(n) * (3+sqrt(8))/2. - Vaclav Kotesovec, Sep 09 2012
The Hankel transform of this sequence gives A006125 = 1, 1, 2, 8, 64, 1024, ...; example: det([1, 1, 3, 11; 1, 3, 11, 45; 3, 11, 45, 197; 11, 45, 197, 903]) = 2^6 = 64. - Philippe Deléham, Mar 02 2004
a(n+1) = Sum_{k=0..floor((n-1)/2)} 2^k * 3^(n-1-2k) * binomial(n-1, 2k) * Catalan(k). This formula counts colored Dyck paths by the same parameter by which Touchard's identity counts ordinary Dyck paths: number of DDUs (U=up step, D=down step). See also Gouyou-Beauchamps reference. - David Callan, Mar 14 2004
From Paul Barry, May 24 2005: (Start)
a(n) = (1/(n+1))*Sum_{k=0..n} C(n+1, k)*C(2n-k, n)*(-1)^k*2^(n-k) [with offset 0].
a(n) = (1/(n+1))*Sum_{k=0..n} C(n+1, k+1)*C(n+k, k)*(-1)^(n-k)*2^k [with offset 0].
a(n) = Sum_{k=0..n} (1/(k+1))*C(n, k)*C(n+k, k)*(-1)^(n-k)*2^k [with offset 0].
a(n) = Sum_{k=0..n} A088617(n, k)*(-1)^(n-k)*2^k [with offset 0]. (End)
E.g.f. of a(n+1) is exp(3*x)*BesselI(1, 2*sqrt(2)*x)/(sqrt(2)*x). - Vladeta Jovovic, Mar 31 2004
Reversion of (x-2*x^2)/(1-x) is g.f. offset 1.
For n>=1, a(n) = Sum_{k=0..n} 2^k*N(n, k) where N(n, k) = (1/n)*C(n, k)*C(n, k+1) are the Narayana numbers (A001263). - Benoit Cloitre, May 10 2003 [This formula counts colored Dyck paths by number of peaks, which is easy to see because the Narayana numbers count Dyck paths by number of peaks and the number of peaks determines the number of interior ascent vertices.]
a(n) = Sum_{k=0..n} A088617(n, k)*2^k*(-1)^(n-k). - Philippe Deléham, Jan 21 2004
For n > 0, a(n) = (1/(n+1)) * Sum_{k = 0 .. n-1} binomial(2*n-k, n) * binomial(n-1, k). This formula counts colored Dyck paths (as above) by number of white vertices. - David Callan, Mar 14 2004
a(n-1) = (d^(n-1)/dx^(n-1))((1-x)/(1-2*x))^n/n!|_{x=0}. (For a proof see the comment on the unsigned row sums of triangle A111785.)
From Wolfdieter Lang, Sep 12 2005: (Start)
a(n) = (1/n)*Sum_{k=1..n} binomial(n, k)*binomial(n+k, k-1).
a(n) = hypergeom([1-n, n+2], [2], -1), n>=1. (End)
a(n) = hypergeom([1-n, -n], [2], 2) for n>=0. - Peter Luschny, Sep 22 2014
a(m+n+1) = Sum_{k>=0} A110440(m, k)*A110440(n, k)*2^k = A110440(m+n, 0). - Philippe Deléham, Sep 14 2005
Sum over partitions formula (reference Schroeder paper p. 362, eq. (1) II). Number the partitions of n according to Abramowitz-Stegun pp. 831-832 (see reference under A105805) with k=1..p(n)= A000041(n). For n>=1: a(n-1) = Sum_{k=2..p(n)} A048996(n,k)*a(1)^e(k, 1)*a(1)^e(k, 2)*...*a(n-2)^e(k, n-1) if the k-th partition of n in the mentioned order is written as (1^e(k, 1), 2^e(k, 2), ..., (n-1)e(k, n-1)). Note that the first (k=1) partition (n^1) has to be omitted. - Wolfdieter Lang, Aug 23 2005
Starting (1, 3, 11, 45, ...), = row sums of triangle A126216 = A001263 * [1, 2, 4, 8, 16, ...]. - Gary W. Adamson, Nov 30 2007
From Paul Barry, May 15 2009: (Start)
G.f.: 1/(1+x-2x/(1+x-2x/(1+x-2x/(1+x-2x/(1-.... (continued fraction).
G.f.: 1/(1-x/(1-x-x/(1-x-x/(1-x-x/(1-... (continued fraction).
G.f.: 1/(1-x-2x^2/(1-3x-2x^2/(1-3x-2x^2/(1-... (continued fraction). (End)
G.f.: 1 / (1 - x / (1 - 2*x / (1 - x / (1 - 2*x / ... )))). - Michael Somos, May 19 2013
a(n) = (LegendreP(n+1,3)-3*LegendreP(n,3))/(4*n) for n>0. - Mark van Hoeij, Jul 12 2010 [This formula is mentioned in S.-J. Kettle's 1982 letter - see link. N. J. A. Sloane, Jun 13 2015]
From Gary W. Adamson, Jul 08 2011: (Start)
a(n) = upper left term in M^n, where M is the production matrix:
  1, 1, 0, 0, 0, 0, ...
  2, 2, 2, 0, 0, 0, ...
  1, 1, 1, 1, 0, 0, ...
  2, 2, 2, 2, 2, 0, ...
  1, 1, 1, 1, 1, 1, ...
  ... (End)
From Gary W. Adamson, Aug 23 2011: (Start)
a(n) is the sum of top row terms of Q^(n-1), where Q is the infinite square production matrix:
  1, 2, 0, 0, 0, ...
  1, 1, 2, 0, 0, ...
  1, 1, 1, 2, 0, ...
  1, 1, 1, 1, 2, ...
  ... (End)
Let h(t) = (1-t)^2/(2*(1-t)^2-1) = 1/(1-(2*t+3*t^2+4*t^3+...)), an o.g.f. for A003480, then for A001003 a(n) = (1/n!)*((h(t)*d/dt)^n) t, evaluated at t=0, with initial n=1. (Cf. A086810.) - Tom Copeland, Sep 06 2011
A006318(n) = 2*a(n) if n>0. - Michael Somos, Mar 31 2007
BINOMIAL transform is A118376 with offset 0. REVERT transform is A153881. INVERT transform is A006318. INVERT transform of A114710. HANKEL transform is A139685. PSUM transform is A104858. - Michael Somos, May 19 2013
G.f.: 1 + x/(Q(0) - x) where Q(k) = 1 + k*(1-x) - x - x*(k+1)*(k+2)/Q(k+1) ; (continued fraction). - Sergei N. Gladkovskii, Mar 14 2013
a(n) = A144944(n,n) = A186826(n,0). - Reinhard Zumkeller, May 11 2013
a(n)=(-1)^n*LegendreP(n,-1,-3)/sqrt(2), n > 0, LegendreP(n,a,b) is the Legendre function. - Karol A. Penson, Jul 06 2013
Integral representation as n-th moment of a positive weight function W(x) = W_a(x) + W_c(x), where W_a(x) = Dirac(x)/2, is the discrete (atomic) part, and W_c(x) = sqrt(8-(x-3)^2)/(4*Pi*x) is the continuous part of W(x) defined on (3 sqrt(8),3+sqrt(8)): a(n) = int( x^n*W_a(x), x=-eps..eps ) + int( x^n*W_c(x), x = 3-sqrt(8)..3+sqrt(8) ), for any eps>0, n>=0. W_c(x) is unimodal, of bounded variation and W_c(3-sqrt(8)) = W_c(3+sqrt(8)) = 0. Note that the position of the Dirac peak (x=0) lies outside support of W_c(x). - Karol A. Penson and Wojciech Mlotkowski, Aug 05 2013
G.f.: 1 + x/G(x) with G(x) = 1 - 3*x - 2*x^2/G(x) (continued fraction). - Nikolaos Pantelidis, Dec 17 2022

A088617 Triangle read by rows: T(n,k) = C(n+k,n)*C(n,k)/(k+1), for n >= 0, k = 0..n.

Original entry on oeis.org

1, 1, 1, 1, 3, 2, 1, 6, 10, 5, 1, 10, 30, 35, 14, 1, 15, 70, 140, 126, 42, 1, 21, 140, 420, 630, 462, 132, 1, 28, 252, 1050, 2310, 2772, 1716, 429, 1, 36, 420, 2310, 6930, 12012, 12012, 6435, 1430, 1, 45, 660, 4620, 18018, 42042, 60060, 51480, 24310, 4862

Offset: 0

N. J. A. Sloane, Nov 23 2003

Row sums: A006318 (Schroeder numbers). Essentially same as triangle A060693 transposed.
T(n,k) is number of Schroeder paths (i.e., consisting of steps U=(1,1), D=(1,-1), H=(2,0) and never going below the x-axis) from (0,0) to (2n,0), having k U's. E.g., T(2,1)=3 because we have UHD, UDH and HUD. - Emeric Deutsch, Dec 06 2003
Little Schroeder numbers A001003 have a(n) = Sum_{k=0..n} A088617(n,k)*(-1)^(n-k)*2^k. - Paul Barry, May 24 2005
Conjecture: The expected number of U's in a Schroeder n-path is asymptotically Sqrt[1/2]*n for large n. - David Callan, Jul 25 2008
T(n, k) is also the number of order-preserving and order-decreasing partial transformations (of an n-chain) of width k (width(alpha) = |Dom(alpha)|). - Abdullahi Umar, Oct 02 2008
The antidiagonals of this lower triangular matrix are the rows of A055151. - Tom Copeland, Jun 17 2015

			Triangle begins:
  [0] 1;
  [1] 1,  1;
  [2] 1,  3,   2;
  [3] 1,  6,  10,    5;
  [4] 1, 10,  30,   35,    14;
  [5] 1, 15,  70,  140,   126,    42;
  [6] 1, 21, 140,  420,   630,   462,   132;
  [7] 1, 28, 252, 1050,  2310,  2772,  1716,   429;
  [8] 1, 36, 420, 2310,  6930, 12012, 12012,  6435,  1430;
  [9] 1, 45, 660, 4620, 18018, 42042, 60060, 51480, 24310, 4862;

Charles Jordan, Calculus of Finite Differences, Chelsea 1965, p. 449.

Michael De Vlieger, Table of n, a(n) for n = 0..11475 (rows 0 <= n <= 150)
Anwar Al Ghabra, K. Gopala Krishna, Patrick Labelle, and Vasilisa Shramchenko, Enumeration of multi-rooted plane trees, arXiv:2301.09765 [math.CO], 2023.
Paul Barry, On Integer-Sequence-Based Constructions of Generalized Pascal Triangles, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.4.
Paul Barry, Continued fractions and transformations of integer sequences, JIS 12 (2009) 09.7.6.
Paul Barry, Generalized Catalan Numbers Associated with a Family of Pascal-like Triangles, J. Int. Seq., Vol. 22 (2019), Article 19.5.8.
Paul Barry, On the inversion of Riordan arrays, arXiv:2101.06713 [math.CO], 2021.
Manosij Ghosh Dastidar and Michael Wallner, Bijections and congruences involving lattice paths and integer compositions, arXiv:2402.17849 [math.CO], 2024. See p. 16.
Samuele Giraudo, Tree series and pattern avoidance in syntax trees, arXiv:1903.00677 [math.CO], 2019.
Hsien-Kuei Hwang and Satoshi Kuriki, Integrated empirical measures and generalizations of classical goodness-of-fit statistics, arXiv:2404.06040 [math.ST], 2024. See p. 11.
C. Jordan, Calculus of Finite Differences, Budapest, 1939. [Annotated scans of pages 448-450 only]
A. Kirillov, On Some Quadratic Algebras I 1/2: Combinatorics of Dunkl and Gaudin Elements, Schubert, Grothendieck, Fuss-Catalan, Universal Tutte and Reduced Polynomials, arXiv preprint arXiv:1502.00426 [math.RT], 2016.
M. Klazar, On numbers of Davenport-Schinzel sequences, Discr. Math., 185 (1998), 77-87.
Paul W. Lapey and Aaron Williams, A Shift Gray Code for Fixed-Content Łukasiewicz Words, Williams College, 2022.
A. Laradji and A. Umar, A. Combinatorial results for semigroups of order-preserving partial transformations, Journal of Algebra 278, (2004), 342-359.
A. Laradji and A. Umar, Combinatorial results for semigroups of order-decreasing partial transformations, J. Integer Seq. 7 (2004), 04.3.8.
Jason P. Smith, The poset of graphs ordered by induced containment, arXiv:1806.01821 [math.CO], 2018.

Diagonals give A000217, A034827, A000910, A088625, A088626, A000108, A001700, A002457, A002802, A002803.
Cf. A000108, A001003, A006318, A060693, A084938, A085478.
Cf. A033282, A055151, A123160.
Cf. A001263, A011973, A055151, A060693, A074909, A086810, A107131.

[[Binomial(n+k,n)*Binomial(n,k)/(k+1): k in [0..n]]: n in [0.. 15]]; // Vincenzo Librandi, Jun 18 2015

R := n -> simplify(hypergeom([-n, n + 1], [2], -x)):
Trow := n -> seq(coeff(R(n, x), x, k), k = 0..n):
seq(print(Trow(n)), n = 0..9); # Peter Luschny, Apr 26 2022

Table[Binomial[n+k, n] Binomial[n, k]/(k+1), {n,0,10}, {k,0,n}]//Flatten (* Michael De Vlieger, Aug 10 2017 *)

{T(n, k)= if(k+1, binomial(n+k, n)*binomial(n, k)/(k+1))}

flatten([[binomial(n+k, 2*k)*catalan_number(k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, May 22 2022

Triangle T(n, k) read by rows; given by [1, 0, 1, 0, 1, 0, 1, 0, 1, 0, ...] DELTA [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...] where DELTA is Deléham's operator defined in A084938.
T(n, k) = A085478(n, k)*A000108(k); A000108 = Catalan numbers. - Philippe Deléham, Dec 05 2003
Sum_{k=0..n} T(n, k)*x^k*(1-x)^(n-k) = A000108(n), A001003(n), A007564(n), A059231(n), A078009(n), A078018(n), A081178(n), A082147(n), A082181(n), A082148(n), A082173(n) for x = 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11. - Philippe Deléham, Aug 18 2005
Sum_{k=0..n} T(n,k)*x^k = (-1)^n*A107841(n), A080243(n), A000007(n), A000012(n), A006318(n), A103210(n), A103211(n), A133305(n), A133306(n), A133307(n), A133308(n), A133309(n) for x = -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8 respectively. - Philippe Deléham, Oct 18 2007
O.g.f. (with initial 1 excluded) is the series reversion with respect to x of (1-t*x)*x/(1+x). Cf. A062991 and A089434. - Peter Bala, Jul 31 2012
G.f.: 1 + (1 - x - T(0))/y, where T(k) = 1 - x*(1+y)/( 1 - x*y/T(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Nov 03 2013
From Peter Bala, Jul 20 2015: (Start)
O.g.f. A(x,t) = ( 1 - x - sqrt((1 - x)^2 - 4*x*t) )/(2*x*t) = 1 + (1 + t)*x + (1 + 3*t + 2*t^2)*x^2 + ....
1 + x*(dA(x,t)/dx)/A(x,t) = 1 + (1 + t)*x + (1 + 4*t + 3*t^2)*x^2 + ... is the o.g.f. for A123160.
For n >= 1, the n-th row polynomial equals (1 + t)/(n+1)*Jacobi_P(n-1,1,1,2*t+1). Removing a factor of 1 + t from the row polynomials gives the row polynomials of A033282. (End)
From Tom Copeland, Jan 22 2016: (Start)
The o.g.f. G(x,t) = {1 - (2t+1) x - sqrt[1 - (2t+1) 2x + x^2]}/2x = (t + t^2) x + (t + 3t^2 + 2t^3) x^2 + (t + 6t^2 + 10t^3 + 5t^3) x^3 + ... generating shifted rows of this entry, excluding the first, was given in my 2008 formulas for A033282 with an o.g.f. f1(x,t) = G(x,t)/(1+t) for A033282. Simple transformations presented there of f1(x,t) are related to A060693 and A001263, the Narayana numbers. See also A086810.
The inverse of G(x,t) is essentially given in A033282 by x1, the inverse of f1(x,t): Ginv(x,t) = x [1/(t+x) - 1/(1+t+x)] = [((1+t) - t) / (t(1+t))] x - [((1+t)^2 - t^2) / (t(1+t))^2] x^2 + [((1+t)^3 - t^3) / (t(1+t))^3] x^3 - ... . The coefficients in t of Ginv(xt,t) are the o.g.f.s of the diagonals of the Pascal triangle A007318 with signed rows and an extra initial column of ones. The numerators give the row o.g.f.s of signed A074909.
Rows of A088617 are shifted columns of A107131, whose reversed rows are the Motzkin polynomials of A055151, related to A011973. The diagonals of A055151 give the rows of A088671, and the antidiagonals (top to bottom) of A088617 give the rows of A107131 and reversed rows of A055151. The diagonals of A107131 give the columns of A055151. The antidiagonals of A088617 (bottom to top) give the rows of A055151.
(End)
T(n, k) = [x^k] hypergeom([-n, 1 + n], [2], -x). - Peter Luschny, Apr 26 2022

A033877 Triangular array read by rows associated with Schroeder numbers: T(1,k) = 1; T(n,k) = 0 if k < n; T(n,k) = T(n,k-1) + T(n-1,k-1) + T(n-1,k).

Original entry on oeis.org

1, 1, 2, 1, 4, 6, 1, 6, 16, 22, 1, 8, 30, 68, 90, 1, 10, 48, 146, 304, 394, 1, 12, 70, 264, 714, 1412, 1806, 1, 14, 96, 430, 1408, 3534, 6752, 8558, 1, 16, 126, 652, 2490, 7432, 17718, 33028, 41586, 1, 18, 160, 938, 4080, 14002, 39152, 89898, 164512, 206098

Offset: 1

N. J. A. Sloane

A106579 is in some ways a better version of this sequence, but since this was entered first it will be the main entry for this triangle.
The diagonals of this triangle are self-convolutions of the main diagonal A006318: 1, 2, 6, 22, 90, 394, 1806, ... . - Philippe Deléham, May 15 2005
From Johannes W. Meijer, Sep 22 2010, Jul 15 2013: (Start)
Note that for the terms T(n,k) of this triangle n indicates the column and k the row.
The triangle sums, see A180662, link Schroeder's triangle with several sequences, see the crossrefs. The mirror of this triangle is A080247.
Quite surprisingly the Kn1p sums, p >= 1, are all related to A026003 and crystal ball sequences for n-dimensional cubic lattices (triangle offset is 0): Kn11(n) = A026003(n), Kn12(n) = A026003(n+2) - 1, Kn13(n) = A026003(n+4) - A005408(n+3), Kn14(n) =  A026003(n+6) - A001844(n+4), Kn15(n) = A026003(n+8) - A001845(n+5), Kn16(n) = A026003(n+10) - A001846(n+6), Kn17(n) = A026003(n+12) - A001847(n+7), Kn18(n) = A026003(n+14) - A001848(n+8), Kn19(n) = A026003(n+16) - A001849(n+9), Kn110(n) = A026003(n+18) - A008417(n+10), Kn111(n) = A026003(n+20) - A008419(n+11), Kn112(n) = A026003(n+22) - A008421(n+12). (End)
T(n,k) is the number of normal semistandard Young tableaux with two columns, one of height k and one of height n. The recursion can be seen by performing jeu de taquin deletion on all instances of the smallest value. (If there are two instances of the smallest value, jeu de taquin deletion will always shorten the right column first and the left column second.) - Jacob Post, Jun 19 2018

			Triangle starts:
  1;
  1,    2;
  1,    4,    6;
  1,    6,   16,   22;
  1,    8,   30,   68,   90;
  1,   10,   48,  146,  304,  394;
  1,   12,   70,  264,  714, 1412, 1806;
  ... - _Joerg Arndt_, Sep 29 2013

T. D. Noe, Rows k = 1..50 of triangle, flattened
Henry Bottomley, Illustration of initial terms
Kevin Brown, Hipparchus on Compound Statements, 1994-2010. - _Johannes W. Meijer_, Sep 22 2010
James East and Nicholas Ham, Lattice paths and submonoids of Z^2, arXiv:1811.05735 [math.CO], 2018.
G. Kreweras, Sur les hiérarchies de segments, Cahiers Bureau Universitaire Recherche Opérationnelle, Cahier 20, Inst. Statistiques, Univ. Paris, 1973.
G. Kreweras, Sur les hiérarchies de segments, Cahiers du Bureau Universitaire de Recherche Opérationnelle, Institut de Statistique, Université de Paris, #20 (1973). (Annotated scanned copy)
G. Kreweras, Aires des chemins surdiagonaux et application à un problème économique, Cahiers du Bureau universitaire de recherche opérationnelle Série Recherche 24 (1976): 1-8. [Annotated scanned copy]
J. W. Meijer, Famous numbers on a chessboard, Acta Nova, Volume 4, No.4, December 2010. pp. 589-598.
J. M. Oh, An explicit formula for the number of fuzzy subgroups of a finite abelian p-group of rank two, Iranian Journal of Fuzzy Systems, Dec 2013, Vol. 10 Issue 6, pp. 125-135.
E. Pergola and R. A. Sulanke, Schroeder Triangles, Paths and Parallelogram Polyominoes, J. Integer Sequences, 1 (1998), #98.1.7.
S. Samieinia, The number of continuous curves in digital geometry, Port. Math. 67 (1) (2010) 75-89, last table.
R. A. Sulanke, Objects counted by the central Delannoy numbers, J. Integer Seq. 6 (2003), Article 03.1.5, 19 pp.
Luis Verde-Star, A Matrix Approach to Generalized Delannoy and Schröder Arrays, J. Int. Seq., Vol. 24 (2021), Article 21.4.1.

Essentially same triangle as A080247 and A080245 but with rows read in reversed order. Also essentially the same triangle as A106579.
Cf. A000007, A006318, A006319, A006320, A006321, A008288, A080243.
Cf. A084938, A103885, A238112, A330801.
Cf. A001003 (row sums), A026003 (antidiagonal sums).
Triangle sums (see the comments): A001003 (Row1, Row2), A026003 (Kn1p, p >= 1), A006603 (Kn21), A227504 (Kn22), A227505 (Kn23), A006603(2*n) (Kn3), A001850 (Kn4), A227506 (Fi1), A010683 (Fi2).

a033877 n k = a033877_tabl !! n !! k
a033877_row n = a033877_tabl !! n
a033877_tabl = iterate
   (\row -> scanl1 (+) $ zipWith (+) ([0] ++ row) (row ++ [0])) [1]
-- Reinhard Zumkeller, Apr 17 2013

function t(n,k)
  if k le 0 or k gt n then return 0;
  elif k eq 1 then return 1;
  else return t(n,k-1) + t(n-1,k-1) + t(n-1,k);
  end if;
end function;
[t(n,k): k in [1..n], n in [1..12]]; // G. C. Greubel, Mar 23 2023

T := proc(n, k) option remember; if n=1 then return(1) fi; if kJohannes W. Meijer, Sep 22 2010, revised Jul 17 2013

Mathematica
```
T[1, ]:= 1; T[n, k_]/;(k
				
```

def A033877_row(n):
    @cached_function
    def prec(n, k):
        if k==n: return 1
        if k==0: return 0
        return prec(n-1,k-1)-2*sum(prec(n,k+i-1) for i in (2..n-k+1))
    return [(-1)^k*prec(n, n-k) for k in (0..n-1)]
for n in (1..10): print(A033877_row(n)) # Peter Luschny, Mar 16 2016

@CachedFunction
def t(n, k): # t = A033847
    if (k<0 or k>n): return 0
    elif (k==1): return 1
    else: return t(n, k-1) + t(n-1, k-1) + t(n-1, k)
flatten([[t(n,k) for k in range(1,n+1)] for n in range(1, 16)]) # G. C. Greubel, Mar 23 2023

As an upper right triangle: a(n, k) = a(n, k-1) + a(n-1, k-1) + a(n-1, k) if k >= n >= 0 and a(n, k) = 0 otherwise.
G.f.: Sum T(n, k)*x^n*y^k = (1-x*y-(x^2*y^2-6*x*y+1)^(1/2)) / (x*(2*y+x*y-1+(x^2*y^2-6*x*y+1)^(1/2))). - Vladeta Jovovic, Feb 16 2003
Another version of A000007 DELTA [0, 2, 1, 2, 1, 2, 1, 2, 1, 2, ...] = 1, 1, 0, 1, 2, 0, 1, 4, 6, 0, 1, 6, 16, 22, 0, 1, ..., where DELTA is Deléham's operator defined in A084938.
Sum_{n=1..floor((k+1)/2)} T(n+p-1, k-n+p) = A026003(2*p+k-3) - A008288(2*p+k-3, p-2), p >= 2, k >= 1. - Johannes W. Meijer, Sep 28 2013
From G. C. Greubel, Mar 23 2023: (Start)
(t(n, k) as a lower triangle)
t(n, k) = t(n, k-1) + t(n-1, k-1) + t(n-1, k) with t(n, 1) = 1.
t(n, n) = A006318(n-1).
t(2*n-1, n) = A330801(n-1).
t(2*n-2, n) = A103885(n-1), n > 1.
Sum_{k=1..n-1} t(n, k) = A238112(n), n > 1.
Sum_{k=1..n} t(n, k) = A001003(n).
Sum_{k=1..n-1} (-1)^(k-1)*t(n, k) = (-1)^n*A001003(n-1), n > 1.
Sum_{k=1..n} (-1)^(k-1)*t(n, k) = A080243(n-1).
Sum_{k=1..floor((n+1)/2)} t(n-k+1, k) = A026003(n-1). (End)

More terms from David W. Wilson

A060693 Triangle (0 <= k <= n) read by rows: T(n, k) is the number of Schröder paths from (0,0) to (2n,0) having k peaks.

Original entry on oeis.org

1, 1, 1, 2, 3, 1, 5, 10, 6, 1, 14, 35, 30, 10, 1, 42, 126, 140, 70, 15, 1, 132, 462, 630, 420, 140, 21, 1, 429, 1716, 2772, 2310, 1050, 252, 28, 1, 1430, 6435, 12012, 12012, 6930, 2310, 420, 36, 1, 4862, 24310, 51480, 60060, 42042, 18018, 4620, 660, 45, 1, 16796

Offset: 0

F. Chapoton, Apr 20 2001

The rows sum to A006318 (Schroeder numbers), the left column is A000108 (Catalan numbers); the next-to-left column is A001700, the alternating sum in each row but the first is 0.
T(n,k) is the number of Schroeder paths (i.e., consisting of steps U=(1,1), D=(1,-1), H=(2,0) and never going below the x-axis) from (0,0) to (2n,0), having k peaks. Example: T(2,1)=3 because we have UU*DD, U*DH and HU*D, the peaks being shown by *. E.g., T(n,k) = binomial(n,k)*binomial(2n-k,n-1)/n for n>0. - Emeric Deutsch, Dec 06 2003
A090181*A007318 as infinite lower triangular matrices. - Philippe Deléham, Oct 14 2008
T(n,k) is also the number of rooted plane trees with maximal degree 3 and k vertices of degree 2 (a node may have at most 2 children, and there are exactly k nodes with 1 child). Equivalently, T(n,k) is the number of syntactically different expressions that can be formed that use a unary operation k times, a binary operation n-k times, and nothing else (sequence of operands is fixed). - Lars Hellstrom (Lars.Hellstrom(AT)residenset.net), Dec 08 2009

			Triangle begins:
00: [    1]
01: [    1,     1]
02: [    2,     3,      1]
03: [    5,    10,      6,      1]
04: [   14,    35,     30,     10,      1]
05: [   42,   126,    140,     70,     15,      1]
06: [  132,   462,    630,    420,    140,     21,     1]
07: [  429,  1716,   2772,   2310,   1050,    252,    28,    1]
08: [ 1430,  6435,  12012,  12012,   6930,   2310,   420,   36,   1]
09: [ 4862, 24310,  51480,  60060,  42042,  18018,  4620,  660,  45,  1]
10: [16796, 92378, 218790, 291720, 240240, 126126, 42042, 8580, 990, 55, 1]
...

Vincenzo Librandi, Rows n = 0..100, flattened
J. Agapito, A. Mestre, P. Petrullo, and M. Torres, Counting noncrossing partitions via Catalan triangles, CEAFEL Seminar, June 30, 2015.
Jean-Christophe Aval and François Bergeron, Rectangular Schröder Parking Functions Combinatorics, arXiv:1603.09487 [math.CO], 2016.
Paul Barry, On Integer-Sequence-Based Constructions of Generalized Pascal Triangles, J. Integer Sequ., Vol. 9 (2006), Article 06.2.4.
Paul Barry, Three Études on a sequence transformation pipeline, arXiv:1803.06408 [math.CO], 2018.
Paul Barry, Riordan Pseudo-Involutions, Continued Fractions and Somos 4 Sequences, arXiv:1807.05794 [math.CO], 2018.
Paul Barry, The Central Coefficients of a Family of Pascal-like Triangles and Colored Lattice Paths, J. Int. Seq., Vol. 22 (2019), Article 19.1.3.
Paul Barry, Generalized Catalan Numbers Associated with a Family of Pascal-like Triangles, J. Int. Seq., Vol. 22 (2019), Article 19.5.8.
Paul Barry, On the inversion of Riordan arrays, arXiv:2101.06713 [math.CO], 2021.
Paul Barry, On Motzkin-Schröder Paths, Riordan Arrays, and Somos-4 Sequences, J. Int. Seq. (2023) Vol. 26, Art. 23.4.7.
Paul Barry, Notes on Riordan arrays and lattice paths, arXiv:2504.09719 [math.CO], 2025. See p. 25.
David Callan and Toufik Mansour, Five subsets of permutations enumerated as weak sorting permutations, arXiv:1602.05182 [math.CO], 2016.
G. E. Cossali, A Common Generating Function of Catalan Numbers and Other Integer Sequences, J. Int. Seqs. 6 (2003).
Dan Drake, Bijections from Weighted Dyck Paths to Schröder Paths, J. Int. Seq. 13 (2010) # 10.9.2
Samuele Giraudo, Tree series and pattern avoidance in syntax trees, arXiv:1903.00677 [math.CO], 2019.
Nate Kube and Frank Ruskey, Sequences That Satisfy a(n-a(n))=0, Journal of Integer Sequences, Vol. 8 (2005), Article 05.5.5.
Krishna Menon and Anurag Singh, Grassmannian permutations avoiding identity, arXiv:2212.13794 [math.CO], 2022.
Jean-Christophe Novelli and Jean-Yves Thibon, Duplicial algebras and Lagrange inversion, arXiv preprint arXiv:1209.5959 [math.CO], 2012.

Cf. A006318, A000108, A001700.
Triangle in A088617 transposed.
Diagonals give A000108, A001700, A002457, A002802, A002803; A000012, A000217, A034827, A000910, A088625, A088626.
T(2n,n) gives A007004.
Cf. A001263, A033282, A086810, A088617, A097610, A101282.

A060693 := (n,k) -> binomial(n,k)*binomial(2*n-k,n)/(n-k+1); # Peter Luschny, May 17 2011

t[n_, k_] := Binomial[n, k]*Binomial[2 n - k, n]/(n - k + 1); Flatten[Table[t[n, k], {n, 0, 9}, {k, 0, n}]] (* Robert G. Wilson v, May 30 2011 *)

T(n, k) = binomial(n, k)*binomial(2*n - k, n)/(n - k + 1);
for(n=0, 10, for(k=0, n, print1(T(n, k),", ")); print); \\ Indranil Ghosh, Jul 28 2017

from sympy import binomial
def T(n, k): return binomial(n, k) * binomial(2 * n - k, n) / (n - k + 1)
for n in range(11): print([T(n, k) for k in range(n + 1)])  # Indranil Ghosh, Jul 28 2017

Triangle T(n, k) (0 <= k <= n) read by rows; given by [1, 1, 1, 1, 1, ...] DELTA [1, 0, 1, 0, 1, 0, ...] where DELTA is the operator defined in A084938. - Philippe Deléham, Aug 12 2003
If C_n(x) is the g.f. of row n of the Narayana numbers (A001263), C_n(x) = Sum_{k=1..n} binomial(n,k-1)*(binomial(n-1,k-1)/k) * x^k and T_n(x) is the g.f. of row n of T(n,k), then T_n(x) = C_n(x+1), or T(n,k) = [x^n]Sum_{k=1..n}(A001263(n,k)*(x+1)^k). - Mitch Harris, Jan 16 2007, Jan 31 2007
G.f.: (1 - t*y - sqrt((1-y*t)^2 - 4*y)) / 2.
T(n, k) = binomial(2n-k, n)*binomial(n, k)/(n-k+1). - Philippe Deléham, Dec 07 2003
A060693(n, k) = binomial(2*n-k, k)*A000108(n-k); A000108: Catalan numbers. - Philippe Deléham, Dec 30 2003
Sum_{k=0..n} T(n,k)*x^k = A000007(n), A000108(n), A006318(n), A047891(n+1), A082298(n), A082301(n), A082302(n), A082305(n), A082366(n), A082367(n), for x = -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, respectively. - Philippe Deléham, Apr 01 2007
T(n,k) = Sum_{j>=0} A090181(n,j)*binomial(j,k). - Philippe Deléham, May 04 2007
Sum_{k=0..n} T(n,k)*x^(n-k) = (-1)^n*A107841(n), A080243(n), A000007(n), A000012(n), A006318(n), A103210(n), A103211(n), A133305(n), A133306(n), A133307(n), A133308(n), A133309(n) for x = -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, respectively. - Philippe Deléham, Oct 18 2007
From Paul Barry, Jan 29 2009: (Start)
G.f.: 1/(1-xy-x/(1-xy-x/(1-xy-x/(1-xy-x/(1-xy-x/(1-.... (continued fraction);
G.f.: 1/(1-(x+xy)/(1-x/(1-(x+xy)/(1-x/(1-(x+xy)/(1-.... (continued fraction). (End)
T(n,k) = [k<=n]*(Sum_{j=0..n} binomial(n,j)^2*binomial(j,k))/(n-k+1). - Paul Barry, May 28 2009
T(n,k) = A104684(n,k)/(n-k+1). - Peter Luschny, May 17 2011
From Tom Copeland, Sep 21 2011: (Start)
With F(x,t) = (1-(2+t)*x-sqrt(1-2*(2+t)*x+(t*x)^2))/(2*x) an o.g.f. (nulling the n=0 term) in x for the A060693 polynomials in t,
  G(x,t) = x/(1+t+(2+t)*x+x^2) is the compositional inverse in x.
Consequently, with H(x,t) = 1/(dG(x,t)/dx) = (1+t+(2+t)*x+x^2)^2 / (1+t-x^2), the n-th A060693 polynomial in t is given by (1/n!)*((H(x,t)*d/dx)^n) x evaluated at x=0, i.e., F(x,t) = exp(x*H(u,t)*d/d) u, evaluated at u = 0.
  Also, dF(x,t)/dx = H(F(x,t),t). (End)
See my 2008 formulas in A033282 to relate this entry to A088617, A001263, A086810, and other matrices. - Tom Copeland, Jan 22 2016
Rows of this entry are non-vanishing antidiagonals of A097610. See p. 14 of Agapito et al. for a bivariate generating function and its inverse. - Tom Copeland, Feb 03 2016
From Werner Schulte, Jan 09 2017: (Start)
T(n,k) = A126216(n,k-1) + A126216(n,k) for 0 < k < n;
Sum_{k=0..n} (-1)^k*(1+x*(n-k))*T(n,k) = x + (1-x)*A000007(n).
(End)
Conjecture: Sum_{k=0..n} (-1)^k*T(n,k)*(n+1-k)^2 = 1+n+n^2. - Werner Schulte, Jan 11 2017

More terms from Vladeta Jovovic, Apr 21 2001
New description from Philippe Deléham, Aug 12 2003
New name using a comment by Emeric Deutsch from Peter Luschny, Jul 26 2017

cp's OEIS Frontend

A001003 Schroeder's second problem (generalized parentheses); also called super-Catalan numbers or little Schroeder numbers.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Haskell

Magma

Maple

Mathematica

PARI

PARI

PARI

PARI

Python

Python

Sage

Formula

A088617 Triangle read by rows: T(n,k) = C(n+k,n)*C(n,k)/(k+1), for n >= 0, k = 0..n.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

SageMath

Formula

A033877 Triangular array read by rows associated with Schroeder numbers: T(1,k) = 1; T(n,k) = 0 if k < n; T(n,k) = T(n,k-1) + T(n-1,k-1) + T(n-1,k).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

Magma

Maple

Mathematica

Sage

SageMath

Formula

Extensions

A060693 Triangle (0 <= k <= n) read by rows: T(n, k) is the number of Schröder paths from (0,0) to (2n,0) having k peaks.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Python

Formula

Extensions

A080245 Inverse of coordination sequence array A113413.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula