A010683 -id:A010683 - cp's OEIS frontend

A227506 Schroeder triangle sums: a(2n-1) = A010683(2n-2) and a(2n) = A010683(2n-1) - A001003(2*n-1).

1, 1, 7, 17, 121, 353, 2591, 8257, 61921, 207905, 1582791, 5501073, 42344121, 150827073, 1170747519, 4247388417, 33186295681, 122125206977, 959260792775, 3570473750929, 28167068630713, 105820555054241, 837838806587167, 3172136074486337

Offset: 1

Johannes W. Meijer, Jul 15 2013

The terms of this sequence equal the Fi1 sums, see A180662, of the Schroeder triangle A033877 (with offset 1 and n for columns and k for rows).

Cf. A033877, A010683, A001003.

A227506 := proc(n) local k, T; T := proc(n, k) option remember; if n=1 then return(1) fi; if kA227506(n), n = 1..24); # Peter Luschny, Jul 17 2013
A227506 := proc(n): if type(n, odd) then A010683(n-1) else A010683(n-1) - A001003(n-1) fi: end: A010683 := proc(n): if n = 0 then 1 else (2/n)*add(binomial(n, k)* binomial(n+k+1, k-1), k=1..n) fi: end: A001003 := proc(n): if n = 0 then 1 else add(binomial(n, j)*binomial(n+j, n-1), j=0..n)/(2*n) fi: end: seq(A227506(n), n=1..24);

T[n_, k_] := T[n, k] = Which[n == 1, 1, k < n, 0, True, T[n, k - 1] + T[n - 1, k - 1] + T[n - 1, k]];
a[n_] := Sum[T[2 k - 1, n], {k, 1, (n + 1)/2}];
Array[a, 24] (* Jean-François Alcover, Jul 11 2019, from Sage *)

def A227506(n):
    @CachedFunction
    def T(n, k):
        if n==1: return 1
        if k A227506(n) for n in (1..24)]  # Peter Luschny, Jul 16 2013

a(n) = Sum_{k=1..floor((n+1)/2)} A033877(2*k-1,n).
a(2*n-1) = A010683(2*n-2) and a(2*n) = A010683(2*n-1) - A001003(2*n-1).
G.f.: (1-4*x+x^2 - sqrt(1-6*x+x^2) + x*sqrt(1+6*x+x^2))/(8*x).

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

Original entry on oeis.org

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

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

A052894 a(n) is the number of Schröder trees on n vertices with a prescribed root.

Original entry on oeis.org

1, 1, 5, 46, 631, 11586, 267369, 7442758, 242833091, 9090987610, 384209125453, 18096001098462, 939991769248239, 53388611049236386, 3291647968944928337, 218948960832551848438, 15629052780600654123739, 1191728692751208814306986, 96675526164560545405689141

Offset: 0

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

Previous name was: A simple grammar.

Number of pointed trees on pointed sets k[1...k...n] with a prescribed point k. - Gus Wiseman, Sep 27 2015

			The a(4) = 46 pointed trees of the form rootpoint[pointedbranch ... pointedbranch] on 1[1 2 3 4] are 1[1 2[2 3[3 4]]], 1[1 2[2 4[3 4]]], 1[1 2[2[2 4] 3]], 1[1 2[2[2 3] 4]], 1[1 2[2 3 4]], 1[1 3[2 3[3 4]]], 1[1 3[2[2 4] 3]], 1[1 3[3 4[2 4]]], 1[1 3[3[2 3] 4]], 1[1 3[2 3 4]], 1[1 4[2 4[3 4]]], 1[1 4[3 4[2 4]]], 1[1 4[2[2 3] 4]], 1[1 4[3[2 3] 4]], 1[1 4[2 3 4]], 1[1[1 3[3 4]] 2], 1[1[1 4[3 4]] 2], 1[1[1[1 4] 3] 2], 1[1[1[1 3] 4] 2], 1[1[1 3 4] 2], 1[1[1 2[2 4]] 3], 1[1[1 4[2 4]] 3], 1[1[1[1 4] 2] 3], 1[1[1[1 2] 4] 3], 1[1[1 2 4] 3], 1[1[1 2[2 3]] 4], 1[1[1 3[2 3]] 4], 1[1[1[1 3] 2] 4], 1[1[1[1 2] 3] 4], 1[1[1 2 3] 4], 1[1[1 2] 3[3 4]], 1[1[1 2] 4[3 4]], 1[1[1 3] 2[2 4]], 1[1[1 3] 4[2 4]], 1[1[1 4] 2[2 3]], 1[1[1 4] 3[2 3]], 1[1 2 3[3 4]], 1[1 2 4[3 4]], 1[1 2[2 4] 3], 1[1 3 4[2 4]], 1[1 2[2 3] 4], 1[1 3[2 3] 4], 1[1[1 4] 2 3], 1[1[1 3] 2 4], 1[1[1 2] 3 4], 1[1 2 3 4]. - _Gus Wiseman_, Sep 27 2015

William Y. C. Chen, A general bijective algorithm for trees, PNAS December 1, 1990 vol. 87 no. 24 9635-9639.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 870

Cf. A053492, A262673, A010683.

spec := [S,{C=Set(B,1 <= card),B=Prod(Z,S),S=Sequence(C)},labeled]: seq(combstruct[count](spec,size=n), n=0..20);

m = 20; (* number of terms *)
Rest@CoefficientList[InverseSeries[Series[2*x - x*E^x, {x, 0, m}], x], x]*Range[0, m-1]! (* Jean-François Alcover, Oct 11 2022 *)

{a(n) = local(A=1); A = (1/x)*serreverse(2*x - x*exp(x +x^2*O(x^n) )); n!*polcoeff(A, n)}
for(n=0, 20, print1(a(n), ", ")) \\ Paul D. Hanna, Jun 19 2015

{Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D}
{a(n)=local(A=1); A = 1 + (1/x)*sum(m=1, n+1, Dx(m-1, (exp(x +x*O(x^n)) - 1)^m * x^m/m!)); n!*polcoeff(A, n)}
for(n=0, 25, print1(a(n), ", ")) \\ Paul D. Hanna, Jun 19 2015

{Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D}
{a(n)=local(A=1+x+x*O(x^n)); A = exp(sum(m=1, n+1, Dx(m-1, (exp(x +x*O(x^n)) - 1)^m * x^(m-1)/m!)+x*O(x^n))); n!*polcoeff(A, n)}
for(n=0, 25, print1(a(n), ", ")) \\ Paul D. Hanna, Jun 19 2015

E.g.f.: RootOf(-2*_Z + _Z*exp(x*_Z) + 1).
a(n) = A053492(n)/n.
From Paul D. Hanna, Jun 19 2015: (Start)
E.g.f. A(x) satisfies:
(1) A(x) = (1/x) * Series_Reversion( 2*x - x*exp(x) ).
(2) A(x) = 1 + (1/x) * Sum_{n>=1} d^(n-1)/dx^(n-1) (exp(x)-1)^n * x^n / n!.
(3) A(x) = exp( Sum_{n>=1} d^(n-1)/dx^(n-1) (exp(x)-1)^n * x^(n-1) / n! ).
(End)
(4) A(x) = Sum_{n>=0} exp(n*x*A(x)) / 2^(n+1). - Paul D. Hanna, Apr 07 2018
a(n) = (1/(n+1)!) * Sum_{k=0..n} (n+k)! * Stirling2(n,k). - Seiichi Manyama, Nov 06 2023

New name from Vaclav Kotesovec, Feb 16 2015

A011117 Triangle of numbers S(x,y) = number of lattice paths from (0,0) to (x,y) that use step set { (0,1), (1,0), (2,0), (3,0), ....} and never pass below y = x.

Original entry on oeis.org

1, 1, 1, 1, 2, 3, 1, 3, 7, 11, 1, 4, 12, 28, 45, 1, 5, 18, 52, 121, 197, 1, 6, 25, 84, 237, 550, 903, 1, 7, 33, 125, 403, 1119, 2591, 4279, 1, 8, 42, 176, 630, 1976, 5424, 12536, 20793, 1, 9, 52, 238, 930, 3206, 9860, 26832, 61921, 103049, 1, 10, 63

Offset: 0

Robert Sulanke (sulanke(AT)diamond.idbsu.edu)

When seen as polynomials with descending coefficients: evaluations are A006318 (x=1), A001003 (x=2).
Triangular array in A104219 transposed. - Philippe Deléham, Mar 16 2005
Triangle T(n,k), 0 <= k <= n, defined by: T(0,0) = 1, T(n,k) = T(n-1,k) + Sum_{j=0..k-1} 2^j*T(n-1,k-1-j). - Philippe Deléham, Oct 10 2005

			Triangle starts:
[0] [1]
[1] [1, 1]
[2] [1, 2,  3]
[3] [1, 3,  7,  11]
[4] [1, 4, 12,  28,  45]
[5] [1, 5, 18,  52, 121,  197]
[6] [1, 6, 25,  84, 237,  550,  903]
[7] [1, 7, 33, 125, 403, 1119, 2591,  4279]
[8] [1, 8, 42, 176, 630, 1976, 5424, 12536, 20793]
[9] [1, 9, 52, 238, 930, 3206, 9860, 26832, 61921, 103049]

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.
E. Pergola and R. A. Sulanke, Schroeder Triangles, Paths and Parallelogram Polyominoes, J. Integer Sequences, 1 (1998), #98.1.7.

Cf. A084938.

Right-hand columns show convolutions of little Schroeder numbers with themselves: A001003, A010683, A010736, A010849.

f[ x_, y_ ] := f[ x, y ] = Module[ {return}, If[ x == 0, return = 1, If[ y == x-1, return = 0, return = f[ x, y-1 ] + Sum[ f[ k, y ], {k, 0, x-1} ] ] ]; return ]; Do[ Print[ Table[ f[ k, j ], {k, 0, j} ] ], {j, 10, 0, -1} ]

def A011117_row(n):
    @cached_function
    def prec(n, k):
        if k==n: return 1
        if k==0: return 0
        return prec(n-1,k-1)+sum(prec(n,k+i-1) for i in (2..n-k+1))
    return [prec(n, n-k) for k in (0..n-1)]
for n in (1..9): print(A011117_row(n)) # Peter Luschny, Mar 16 2016

S(m, n) = ((n-m+1)/(n+1))*Sum_{i=0..m-1} 2^(m-i-1)*binomial(n+1, i+1)*binomial(m-1, i).
Another version of triangle [1, 0, 0, 0, 0, 0, ...] DELTA [0, 1, 2, 1, 2, 1, 2, 1, 2, 1, ...] = 1, 1, 0, 1, 1, 0, 1, 2, 3, 0, 1, 3, 7, 11, 0, 1, 4, 12, 28, 45, 0, 1, ..., where DELTA is Deléham's operator defined in A084938.
G.f.: 2/(1 + uv - 2v + sqrt(1 - 6uv + u^2v^2)). - Emeric Deutsch, Dec 25 2003
Sum_{k = 0..n} T(n, k) = A006318(n), large Schroeder numbers. - Philippe Deléham, Jul 10 2004. (This is because T(n, k) = number of royal paths (A006318) of length n with exactly n-k Northeast steps lying on the line y=x. - David Callan, Aug 02 2004)
S(n,m) = ((n-m+1)/m)*Sum_{k=1..m} binomial(m,k)*binomial(n+k,k-1), n >= m > 1; S(n,0)=1; S(n,m)=0, n < m. See the corresponding formula for A104219. - Wolfdieter Lang, Mar 16 2009

A186826 Riordan array (s(x),x*S(x)) where s(x) is the g.f. of the little Schroeder numbers A001003, and S(x) is the g.f. of the large Schroeder numbers A006318.

Original entry on oeis.org

1, 1, 1, 3, 3, 1, 11, 11, 5, 1, 45, 45, 23, 7, 1, 197, 197, 107, 39, 9, 1, 903, 903, 509, 205, 59, 11, 1, 4279, 4279, 2473, 1061, 347, 83, 13, 1, 20793, 20793, 12235, 5483, 1949, 541, 111, 15, 1, 103049, 103049, 61463, 28435, 10717, 3285, 795, 143, 17, 1, 518859, 518859, 312761, 148249, 58351, 19199, 5197, 1117, 179, 19, 1

Offset: 0

Paul Barry, Feb 27 2011

Reverse of A144944. Inverse of A186827.

			Triangle begins
       1;
       1,      1;
       3,      3,      1;
      11,     11,      5,      1;
      45,     45,     23,      7,     1;
     197,    197,    107,     39,     9,     1;
     903,    903,    509,    205,    59,    11,    1;
    4279,   4279,   2473,   1061,   347,    83,   13,    1;
   20793,  20793,  12235,   5483,  1949,   541,  111,   15,   1;
  103049, 103049,  61463,  28435, 10717,  3285,  795,  143,  17,  1;
  518859, 518859, 312761, 148249, 58351, 19199, 5197, 1117, 179, 19, 1;
Production matrix of this triangle begins
  1, 1;
  2, 2, 1;
  2, 2, 2, 1;
  2, 2, 2, 2, 1;
  2, 2, 2, 2, 2, 1;
  2, 2, 2, 2, 2, 2, 1;
  2, 2, 2, 2, 2, 2, 2, 1;
  2, 2, 2, 2, 2, 2, 2, 2, 1;
  2, 2, 2, 2, 2, 2, 2, 2, 2, 1;
For instance, 107=1*45+2*23+2*7+2*1.

Reinhard Zumkeller, Rows n = 0..125 of triangle, flattened

Cf. A001003, A006318, A010683 (row sums), A144944 (row reverse), A186827 (inverse), A186828 (diagonal sums), A239204.

a186826 n k = a186826_tabl !! n !! k
a186826_row n = a186826_tabl !! n
a186826_tabl = map reverse a144944_tabl
-- Reinhard Zumkeller, May 11 2013

t[, 0]=1; t[p, p_]:= t[p, p]= t[p, p-1]; t[p_, q_]:= t[p, q]= t[p, q -1] + t[p-1, q] + t[p-1, q-1];
Table[t[p, q], {p,0,10}, {q,p,0,-1}]//Flatten (* Jean-François Alcover, Jul 16 2019 *)

@CachedFunction
def t(n,k):
    if (k<0 or k>n): return 0
    elif (k==0): return 1
    elif (kA186826(n,k): return t(n+2,n-k)
flatten([[A186826(n,k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Mar 11 2023

Riordan array ((1+x+sqrt(1-6*x+x^2))/(4*x), (1-x-sqrt(1-6*x+x^2))/2).
Sum_{k=0..n} T(n,k) = A010683(n).
Sum_{k=0..floor(n/2)} T(n-k, k) = A186828(n).
R(n,k) = k*Sum_{i=0..n-k} (A001003(i)/(n-i))*Sum_{m=0..n-k-i} binomial(n-i,m)*binomial(2*(n-i)-m-k-1, n-i-1), k>0, R(n,0) = A001003(n). - Vladimir Kruchinin, Mar 09 2011
Sum_{k=0..n} (-1)^k*T(n, k) = A239204(n-2). - G. C. Greubel, Mar 11 2023

A144944 Super-Catalan triangle (read by rows) = triangular array associated with little Schroeder numbers (read by rows): T(0,0)=1, T(p,q) = T(p,q-1) if 0 < p = q, T(p,q) = T(p,q-1) + T(p-1,q) + T(p-1,q-1) if -1 < p < q and T(p,q) = 0 otherwise.

Original entry on oeis.org

1, 1, 1, 1, 3, 3, 1, 5, 11, 11, 1, 7, 23, 45, 45, 1, 9, 39, 107, 197, 197, 1, 11, 59, 205, 509, 903, 903, 1, 13, 83, 347, 1061, 2473, 4279, 4279, 1, 15, 111, 541, 1949, 5483, 12235, 20793, 20793, 1, 17, 143, 795, 3285, 10717, 28435, 61463, 103049, 103049

Offset: 0

Johannes Fischer (Fischer(AT)informatik.uni-tuebingen.de), Sep 26 2008

			First few rows of the triangle:
  1
  1,  1
  1,  3,  3
  1,  5, 11,  11
  1,  7, 23,  45,  45
  1,  9, 39, 107, 197, 197
  1, 11, 59, 205, 509, 903, 903

Reinhard Zumkeller, Rows n = 0..125 of triangle, flattened
Johannes Fischer and Volker Heun, Finding Range Minima in the Middle: Approximations and Applications, Math. Comput. Sci. 3(1): 17-30 (2010). See Eq. (3.3)
Andrew Misseldine, Counting Schur Rings over Cyclic Groups, arXiv preprint arXiv:1508.03757 [math.RA], 2015. See Fig. 8.

Super-Catalan numbers or little Schroeder numbers (cf. A001003) appear on the diagonal.
Generalizes the Catalan triangle (A009766) and hence the ballot Numbers.
Cf. A033877 for a similar triangle derived from the large Schroeder numbers (A006318).
Cf. A010683 (row sums), A186826 (rows reversed).
Cf. A239204, A247623.

a144944 n k = a144944_tabl !! n !! k
a144944_row n = a144944_tabl !! n
a144944_tabl = iterate f [1] where
   f us = vs ++ [last vs] where
     vs = scanl1 (+) $ zipWith (+) us $ [0] ++ us
-- Reinhard Zumkeller, May 11 2013

t[, 0]=1; t[p, p_]:= t[p, p]= t[p, p-1]; t[p_, q_]:= t[p, q]= t[p, q-1] + t[p-1, q] + t[p-1, q-1]; Flatten[Table[ t[p, q], {p,0,6}, {q,0, p}]] (* Jean-François Alcover, Dec 19 2011 *)

@CachedFunction
def t(n,k):
    if (k<0 or k>n): return 0
    elif (k==0): return 1
    elif (kG. C. Greubel, Mar 11 2023

From G. C. Greubel, Mar 11 2023: (Start)
Sum_{k=0..n} T(n, k) = A010683(n).
Sum_{k=0..n} (-1)^k*T(n, k) = A239204(n-2).
Sum_{k=0..floor(n/2)} T(n-k, k) = A247623(n). (End)

A177010 G.f.: ((1-q)^2+(1+q)sqrt(1-6q+q^2))/2.

Original entry on oeis.org

1, -2, -3, -8, -28, -112, -484, -2200, -10364, -50144, -247684, -1243816, -6331164, -32591184, -169376484, -887465784, -4682990076, -24864759744, -132745182724, -712136879944, -3837043171100, -20755343638064, -112668274522852, -613581530620888, -3351355226348668, -18354390999804832, -100771410237846404, -554536090460808680

Offset: 0

N. J. A. Sloane, Dec 08 2010

sign

Harris Kwong, On recurrences of Fahr and Ringel: an alternate approach, Fibonacci Quart. 48 (2010), no. 4, 363-365; see p. 364.

Cf. A010683 (a closely-related g.f.)

CoefficientList[Series[((1 - x)^2 + (1 + x) Sqrt[1 - 6 x + x^2])/2, {x, 0, 27}], x] (* Michael De Vlieger, Oct 12 2016 *)

A010849 Let S(x,y) = number of lattice paths from (0,0) to (x,y) that use the step set { (0,1), (1,0), (2,0), (3,0), ....} and never pass below y = x. Sequence gives S(n-3,n).

Original entry on oeis.org

1, 4, 18, 84, 403, 1976, 9860, 49912, 255701, 1323292, 6907830, 36331500, 192339687, 1024140336, 5481165832, 29469454640, 159094662121, 862087135988, 4687164401114, 25562520325828, 139803777476859, 766578879858024

Offset: 0

Robert Sulanke (sulanke(AT)diamond.idbsu.edu)

nonn

Number of dissections of a convex polygon with n+5 sides that have a pentagon over a fixed side (the base) of the polygon. Example: a(1)=4 because the only dissections of the convex hexagon ABCDEF (AB being the base), that have a pentagon over AB are the dissections made by the diagonals FD, EC, AE and BD, respectively. - Emeric Deutsch, Dec 27 2003

a(n-1) = number of royal paths (A006318) from (0,0) to (n,n) with exactly 3 diagonal steps on the line y=x. - David Callan, Jul 15 2004

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Cf. A001003.
Right-hand column 4 of triangle A011117.
Fourth column of convolution triangle A011117.

f[ x_, y_ ] := f[ x, y ] = Module[ {return}, If[ x == 0, return = 1, If[ y == x-1, return = 0, return = f[ x, y-1 ] + Sum[ f[ k, y ], {k, 0, x-1} ] ] ]; return ]; Do[ Print[ Table[ f[ k, j ], {k, 0, j} ] ], {j, 10, 0, -1} ] (* End *)
CoefficientList[Series[(1 + x - Sqrt[1 - 6 x + x^2])^4 / (256 x^4), {x, 0, 30}], x] (* Vincenzo Librandi, May 03 2013 *)

x='x+O('x^66); Vec((1+x-sqrt(1-6*x+x^2))^4/(256*x^4)) \\ Joerg Arndt, May 04 2013

G.f.: (1+z-sqrt(1-6*z+z^2))^4/(256*z^4). 4-fold convolution of A001003 with itself. Convolution of A010683 with itself. - Emeric Deutsch, Dec 27 2003
a(n) = (4/n)*sum(binomial(n, k)*binomial(n+k+3, k-1), k=1..n) = 4*hypergeom([1-n, n+5], [2], -1), n>=1, a(0)=1.
Recurrence: n*(n+4)*a(n) = (7*n^2+16*n-3)*a(n-1) - (7*n^2-2*n-12)*a(n-2) + (n-3)*(n+1)*a(n-3). - Vaclav Kotesovec, Oct 07 2012
a(n) ~ sqrt(1632+1154*sqrt(2))*(3+2*sqrt(2))^n/(4*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Oct 07 2012
Recurrence (an alternative): (n+4)*a(n)  = (8-n)*a(n-8) + 4*(2*n-13)*a(n-7) + 12*(5-n)*a(n-6) + 4*(7-2*n)*a(n-5) + 26*(n-2)*a(n-4) + 4*(1-2*n)*a(n-3)  - 12*(n+1)*a(n-2) + 4*(2*n+5)*a(n-1), n>=8. - Fung Lam, Feb 18 2014

More terms from Emeric Deutsch, Dec 27 2003

A091370 Triangle read by rows: T(n,k) is the number of dissections of a convex n-gon by nonintersecting diagonals, having a k-gon over a fixed edge (base).

Original entry on oeis.org

1, 2, 1, 7, 3, 1, 28, 12, 4, 1, 121, 52, 18, 5, 1, 550, 237, 84, 25, 6, 1, 2591, 1119, 403, 125, 33, 7, 1, 12536, 5424, 1976, 630, 176, 42, 8, 1, 61921, 26832, 9860, 3206, 930, 238, 52, 9, 1, 310954, 134913, 49912, 16470, 4908, 1316, 312, 63, 10, 1

Offset: 3

Emeric Deutsch, Mar 01 2004

Row sums give the little Schroeder numbers (A001003). Column 3 (first column, corresponding to k=3) gives A010683.

Number of short bushes (i.e. ordered trees with no vertices of outdegree 1) with n-1 leaves and having root of degree k-1. Example: T(5,3)=7 because, in addition to the five binary trees with 6 edges we have (i) two edges rb, rc hanging from the root r with three edges hanging from vertex b and (ii) two edges rb, rc hanging from the root r with three edges hanging from vertex c.

			T(5,4)=3 because the dissections of the pentagon ABCDEA that have a quadrilateral over the base AB are obtained by the diagonals (i) CE, (ii) AD and (iii) BD, respectively.
Triangle starts:
1;
2,1;
7,3,1;
28,12,4,1;
121,52,18,5,1;
...

P. Flajolet and M. Noy, Analytic combinatorics of non-crossing configurations, Discrete Math., 204, 203-229, 1999.
J.-C. Novelli and J.-Y. Thibon, Noncommutative Symmetric Functions and Lagrange Inversion, arXiv:math/0512570 [math.CO], 2005-2006.

Cf. A001003, A010683.

a := proc(n,k) if k=0 or k=1 or k=2 then 0 elif k=n then 1 elif k hypergeom([1-N, N+2], [2], -1);
f := n -> 1+add(simplify(c(i))*x^i,i=1..n):
s := j -> coeff(series(f(j)^2/(1-x*t*f(j)),x,j+1),x,j):
seq(coeff(s(n),t,j),j=0..n) end:
seq(T_row(n), n=0..9); # Peter Luschny, Oct 30 2015

T[n_, n_] = 1; T[n_, k_] := (k - 1)/(n - k)*Sum[2^j*Binomial[n - 2, n - k - 1 - j]*Binomial[n - k, j], {j, 0, n - k - 1}];
Table[T[n, k], {n, 3, 13}, {k, 3, n}] // Flatten (* Jean-François Alcover, Nov 24 2017 *)

T(n, k) = [(k-1)/(n-k)]*sum(2^j*binomial(n-2, n-k-1-j)*binomial(n-k, j), j=0..n-k-1).

G.f.: t^3*z^3*S^2/(1-t*z*S), where S = (1+z-sqrt(1-6*z+z^2))/(4*z) is the g.f. of the little Schroeder numbers (A001003).

cp's OEIS Frontend

A227506 Schroeder triangle sums: a(2*n-1) = A010683(2*n-2) and a(2*n) = A010683(2*n-1) - A001003(2*n-1).

Views

Author

Keywords

Comments

Crossrefs

Programs

Maple

Mathematica

Sage

Formula

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

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

A052894 a(n) is the number of Schröder trees on n vertices with a prescribed root.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

PARI

PARI

Formula

Extensions

A011117 Triangle of numbers S(x,y) = number of lattice paths from (0,0) to (x,y) that use step set { (0,1), (1,0), (2,0), (3,0), ....} and never pass below y = x.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Sage

Formula

A186826 Riordan array (s(x),x*S(x)) where s(x) is the g.f. of the little Schroeder numbers A001003, and S(x) is the g.f. of the large Schroeder numbers A006318.

A227506 Schroeder triangle sums: a(2n-1) = A010683(2n-2) and a(2n) = A010683(2n-1) - A001003(2*n-1).

A177010 G.f.: ((1-q)^2+(1+q)sqrt(1-6q+q^2))/2.