A033321 -id:A033321 - cp's OEIS frontend

A007317 Binomial transform of Catalan numbers.

1, 2, 5, 15, 51, 188, 731, 2950, 12235, 51822, 223191, 974427, 4302645, 19181100, 86211885, 390248055, 1777495635, 8140539950, 37463689775, 173164232965, 803539474345, 3741930523740, 17481709707825, 81912506777200, 384847173838501, 1812610804416698

Offset: 1

N. J. A. Sloane, Mira Bernstein

Partial sums of A002212 (the restricted hexagonal polyominoes with n cells). 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-2,0), with no peaks at even level. Example: a(3)=5 because among the six Schroeder paths from (0,0) to (4,0) only UUDD has a peak at an even level. - Emeric Deutsch, Dec 06 2003
Number of binary trees of weight n where leaves have positive integer weights. Non-commutative Non-associative version of partitions of n. - Michael Somos, May 23 2005
Appears also as the number of Euler trees with total weight n (associated with even switching class of matrices of order 2n). - David Garber, Sep 19 2005
Number of symmetric hex trees with 2n-1 edges; also number of symmetric hex trees with 2n-2 edges. A hex tree is a rooted tree where each vertex has 0, 1, or 2 children and, when only one child is present, it is either a left child, or a median child, or a right child (name due to an obvious bijection with certain tree-like polyhexes; see the Harary-Read reference). A hex tree is symmetric if it is identical with its reflection in a bisector through the root. - Emeric Deutsch, Dec 19 2006
The Hankel transform of [1, 2, 5, 15, 51, 188, ...] is [1, 1, 1, 1, 1, ...], see A000012 ; the Hankel transform of [2, 5, 15, 51, 188, 731, ...] is [2, 5, 13, 34, 89, ...], see A001519. - Philippe Deléham, Dec 19 2006
a(n) = number of 321-avoiding partitions of [n]. A partition is 321-avoiding if the permutation obtained from its canonical form (entries in each block listed in increasing order and blocks listed in increasing order of their first entries) is 321-avoiding. For example, the only partition of [5] that fails to be 321-avoiding is 15/24/3 because the entries 5,4,3 in the permutation 15243 form a 321 pattern. - David Callan, Jul 22 2008
The sequence 1,1,2,5,15,51,188,... has Hankel transform A001519. - Paul Barry, Jan 13 2009
From Gary W. Adamson, May 17 2009: (Start)
  Equals INVERT transform of A033321: (1, 1, 2, 6, 21, 79, 311, ...).
  Equals INVERTi transform of A002212: (1, 3, 10, 36, 137, ...).
  Convolved with A026378, (1, 4, 17, 75, 339, ...) = A026376: (1, 6, 30, 144, ...)
(End)
a(n) is the number of vertices of the composihedron CK(n). The composihedra are a sequence of convex polytopes used to define maps of certain homotopy H-spaces. They are cellular quotients of the multiplihedra and cellular covers of the cubes. - Stefan Forcey (sforcey(AT)gmail.com), Dec 17 2009
a(n) is the number of Motzkin paths of length n-1 in which the (1,0)-steps at level 0 come in 2 colors and those at a higher level come in 3 colors. Example: a(4)=15 because we have 2^3 = 8 paths of shape UHD, 2 paths of shape HUD, 2 paths of shape UDH, and 3 paths of shape UHD; here U=(1,1), H=(1,0), and D=(1,-1). - Emeric Deutsch, May 02 2011
REVERT transform of (1, 2, -3, 5, -8, 13, -21, 34, ... ) where the entries are Fibonacci numbers, A000045. Equivalently, coefficients in the series reversion of x(1-x)/(1+x-x^2). This means that the substitution of the gf (1-x-(1-6x+5x^2)^(1/2))/(2(1-x)) for x in x(1-x)/(1+x-x^2) will simplify to x. - David Callan, Nov 11 2012
The number of plane trees with nodes that have positive integer weights and whose total weight is n. - Brad R. Jones, Jun 12 2014
From Tom Copeland, Nov 02 2014: (Start)
Let P(x) = x/(1+x) with comp. inverse Pinv(x) = x/(1-x) = -P[-x], and C(x)= [1-sqrt(1-4x)]/2, an o.g.f. for the shifted Catalan numbers A000108, with inverse Cinv(x) = x * (1-x).
Fin(x) = P[C(x)] = C(x)/[1 + C(x)] is an o.g.f. for the Fine numbers, A000957 with inverse Fin^(-1)(x) = Cinv[Pinv(x)] = Cinv[-P(-x)].
Mot(x) = C[P(x)] = C[-Pinv(-x)] gives an o.g.f. for shifted A005043, the Motzkin or Riordan numbers with comp. inverse Mot^(-1)(x) = Pinv[Cinv(x)] = (x - x^2) / (1 - x + x^2) (cf. A057078).
BTC(x) = C[Pinv(x)] gives A007317, a binomial transform of the Catalan numbers, with BTC^(-1)(x) = P[Cinv(x)] = (x-x^2) / (1 + x - x^2).
Fib(x) = -Fin[Cinv(Cinv(-x))] = -P[Cinv(-x)] = x + 2 x^2 + 3 x^3 + 5 x^4 + ... = (x+x^2)/[1-x-x^2] is an o.g.f. for the shifted Fibonacci sequence A000045, so the comp. inverse is Fib^(-1)(x) = -C[Pinv(-x)] = -BTC(-x) and Fib(x) = -BTC^(-1)(-x).
Generalizing to P(x,t) = x /(1 + t*x) and Pinv(x,t) = x /(1 - t*x) = -P(-x,t) gives other relations to lattice paths, such as the o.g.f. for A091867, C[P[x,1-t]], and that for A104597, Pinv[Cinv(x),t+1].
(End)
Starting with offset 0, a(n) is also the number of Schröder paths of semilength n avoiding UH (an up step directly followed by a long horizontal step). Example: a(2)=5 because among the six possible Schröder paths of semilength 2 only UHD contains UH. - Valerie Roitner, Jul 23 2020

			a(3)=5 since {3, (1+2), (1+(1+1)), (2+1), ((1+1)+1)} are the five weighted binary trees of weight 3.
G.f. = x + 2*x^2 + 5*x^3 + 15*x^4 + 51*x^5 + 188*x^6 + 731*x^7 + 2950*x^8 + 12235*x^9 + ... _Michael Somos_, Jan 17 2018

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Christopher Bao, Yunseo Choi, Katelyn Gan, and Owen Zhang, On a Conjecture by Baril, Cerbai, Khalil, and Vajnovszki on Two Restricted Stacks, arXiv:2308.09344 [math.CO], 2023.
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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], 2015-2017.
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Giulio Cerbai, Anders Claesson, Luca Ferrari, and Einar Steingrímsson, Sorting with pattern-avoiding stacks: the 132-machine, arXiv:2006.05692 [math.CO], 2020.
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S. J. Cyvin et al., Enumeration and classification of benzenoid systems. 32. Normal perifusenes with two internal vertices, J. Chem. Inform. Comput. Sci., 32 (1992), 532-540.
S. J. Cyvin et al., Graph-theoretical studies on fluoranthenoids and fluorenoids:enumeration of some catacondensed systems, J. Molec. Struct. (Theochem), 285 (1993), 179-185.
S. J. Cyvin et al., Enumeration and Classification of Certain Polygonal Systems Representing Polycyclic Conjugated Hydrocarbons: Annelated Catafusenes, J. Chem. Inform. Comput. Sci., 34 (1994), 1174-1180.
Dennis E. Davenport, Louis W. Shapiro, and Leon C. Woodson, The Double Riordan Group, The Electronic Journal of Combinatorics, 18(2) (2012), #P33. - From _N. J. A. Sloane_, May 11 2012
Isaac DeJager, Madeleine Naquin, and Frank Seidl, Colored Motzkin Paths of Higher Order, VERUM 2019.
Rui Duarte and António Guedes de Oliveira, Generating functions of lattice paths, Univ. do Porto (Portugal 2023).
Paul Duncan and Einar Steingrimsson, Pattern avoidance in ascent sequences, arXiv preprint arXiv:1109.3641 [math.CO], 2011.
Francesc Fite, Kiran S. Kedlaya, Victor Rotger, and Andrew V. Sutherland, Sato-Tate distributions and Galois endomorphism modules in genus 2, arXiv preprint arXiv:1110.6638 [math.NT], 2011-2012.
S. Forcey, Quotients of the multiplihedron as categorified associahedra,Homotopy, Homology and Applications, vol. 10(2), 227-256, 2008. [From Stefan Forcey (sforcey(AT)gmail.com), Dec 17 2009]
Ira M. Gessel and Jang Soo Kim, A note on 2-distant noncrossing partitions and weighted Motzkin paths, arXiv:1003.5301 [math.CO], 2010.
Ira M. Gessel and Jang Soo Kim, A note on 2-distant noncrossing partitions and weighted Motzkin paths, Discrete Math. 310 (2010), no. 23, 3421--3425. MR2721104 (2011j:05350). See Eq. (1). - _N. J. A. Sloane_, Jul 05 2014
Juan B. Gil and Jordan O. Tirrell, A simple bijection for classical and enhanced k-noncrossing partitions, arXiv:1806.09065 [math.CO], 2018. Also Discrete Mathematics (2019) Article 111705. doi:10.1016/j.disc.2019.111705
Juan B. Gil and Michael D. Weiner, On pattern-avoiding Fishburn permutations, arXiv:1812.01682 [math.CO], 2018.
Ricardo Gómez Aíza, Trees with flowers: A catalog of integer partition and integer composition trees with their asymptotic analysis, arXiv:2402.16111 [math.CO], 2024. See pp. 18-19.
U. Grude, Java ist eine Sprache: Rekursive Unterprogramme. See page 4.
Nancy S. S. Gu, Nelson Y. Li, and Toufik Mansour, 2-Binary trees: bijections and related issues, Discr. Math., 308 (2008), 1209-1221.
Frank Harary and Ronald C. Read, The enumeration of tree-like polyhexes, Proc. Edinburgh Math. Soc. (2) 17 (1970), 1-13.
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.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 124
Bradley Robert Jones, On tree hook length formulas, Feynman rules and B-series, Master's thesis, Simon Fraser University, 2014.
Hana Kim and R. P. Stanley, A refined enumeration of hex trees and related polynomials, Preprint 2015.
Jang Soo Kim, Bijections on two variations of noncrossing partitions, Discrete Math., 311 (2011), 1057-1063.
John W. Layman, The Hankel Transform and Some of its Properties, J. Integer Sequences, 4 (2001), #01.1.5.
Toufik Mansour and Simone Severini, Enumeration of (k,2)-noncrossing partitions, Discrete Math., 308 (2008), 4570-4577.
Toufik Mansour and Mark Shattuck, Some enumerative results related to ascent sequences, arXiv preprint arXiv:1207.3755 [math.CO], 2012. - From _N. J. A. Sloane_, Dec 22 2012
Igor Pak, Partition identities and geometric bijections, Proc. Amer. Math. Soc. 132 (2004), 3457-3462.
Lara K. Pudwell, Ascent sequences and the binomial convolution of Catalan numbers, arXiv:1408.6823 [math.CO], 2014.
Lara Pudwell and Andrew Baxter, Ascent sequences avoiding pairs of patterns, 2014.
Lara Pudwell, Pattern-avoiding ascent sequences, Slides from a talk, 2015.
Valerie Roitner, The vectorial kernel method for walks with longer steps, arXiv:2008.02240 [math.CO], 2020.
N. J. A. Sloane, Transforms
Paveł Szabłowski, Beta distributions whose moment sequences are related to integer sequences listed in the OEIS, Contrib. Disc. Math. (2024) Vol. 19, No. 4, 85-109. See p. 98.
Makhin Thitsa and W. Steven Gray, On the Radius of Convergence of Interconnected Analytic Nonlinear Input-Output Systems, SIAM Journal on Control and Optimization, Vol. 50, No. 5, 2012, pp. 2786-2813. - From _N. J. A. Sloane_, Dec 26 2012
S. H. F. Yan, Schröder paths and Pattern Avoiding Partitions, Int. J. Contemp. Math. Sciences, Vol. 4, no. 20, pp. 979-986, 2009.

See A181768 for another version. - N. J. A. Sloane, Nov 12 2010
First column of triangle A104259. Row sums of absolute values of A091699.
Number of vertices of multiplihedron A121988.
m-th binomial transform of the Catalan numbers: A126930 (m = -2), A005043 (m = -1), A000108 (m = 0), A064613 (m = 2), A104455 (m = 3), A104498 (m = 4) and A154623 (m = 5).
Cf. A055879, A033321, A026376, A026378, A059346, A000045, A000957, A057078,  A091867, A104597, A249548, A162326.

G := (1-sqrt(1-4*z/(1-z)))*1/2: Gser := series(G, z = 0, 30): seq(coeff(Gser, z, n), n = 1 .. 26); # Emeric Deutsch, Aug 12 2007
seq(round(evalf(JacobiP(n-1,1,-n-1/2,9)/n,99)),n=1..25); # Peter Luschny, Sep 23 2014

Rest@ CoefficientList[ InverseSeries[ Series[(y - y^2)/(1 + y - y^2), {y, 0, 26}], x], x] (* then A(x)=y(x); note that InverseSeries[Series[y-y^2, {y, 0, 24}], x] produces A000108(x) *) (* Len Smiley, Apr 10 2000 *)
Range[0, 25]! CoefficientList[ Series[ Exp[ 3x] (BesselI[0, 2x] - BesselI[1, 2x]), {x, 0, 25}], x] (* Robert G. Wilson v, Apr 15 2011 *)
a[n_] := Sum[ Binomial[n, k]*CatalanNumber[k], {k, 0, n}]; Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Aug 07 2012 *)
Rest[CoefficientList[Series[3/2 - (1/2) Sqrt[(1 - 5 x)/(1 - x)], {x, 0, 40}], x]] (* Vincenzo Librandi, Nov 03 2014 *)
Table[Hypergeometric2F1[1/2, -n+1, 2, -4], {n, 1, 30}] (* Vaclav Kotesovec, May 12 2022 *)

{a(n) = my(A); if( n<2, n>0, A=vector(n); for(j=1,n, A[j] = 1 + sum(k=1,j-1, A[k]*A[j-k])); A[n])}; /* Michael Somos, May 23 2005 */

{a(n) = if( n<1, 0, polcoeff( serreverse( (x - x^2) / (1 + x - x^2) + x * O(x^n)), n))}; /* Michael Somos, May 23 2005 */

/* Offset = 0: */ {a(n)=local(A=1+x);for(i=1,n, A=sum(m=0,n, x^m*sum(k=0,m,A^k)+x*O(x^n))); polcoeff(A,n)} \\ Paul D. Hanna

(n+2)*a(n+2) = (6n+4)*a(n+1) - 5n*a(n).
G.f.: 3/2-(1/2)*sqrt((1-5*x)/(1-x)) [Gessel-Kim]. - N. J. A. Sloane, Jul 05 2014
G.f. for sequence doubled: (1/(2*x))*(1+x-(1-x)^(-1)*(1-x^2)^(1/2)*(1-5*x^2)^(1/2)).
a(n) = hypergeom([1/2, -n], [2], -4), n=0, 1, 2...; Integral representation as n-th moment of a positive function on a finite interval of the positive half-axis: a(n)=int(x^n*sqrt((5-x)/(x-1))/(2*Pi), x=1..5), n=0, 1, 2... This representation is unique. - Karol A. Penson, Sep 24 2001
a(1)=1, a(n)=1+sum(i=1, n-1, a(i)*a(n-i)). - Benoit Cloitre, Mar 16 2004
a(n) = Sum_{k=0..n} (-1)^k*3^(n-k)*binomial(n, k)*binomial(k, floor(k/2)) [offset 0]. - Paul Barry, Jan 27 2005
G.f. A(x) satisfies 0=f(x, A(x)) where f(x, y)=x-(1-x)(y-y^2). - Michael Somos, May 23 2005
G.f. A(x) satisfies 0=f(x, A(x), A(A(x))) where f(x, y, z)=x(z-z^2)+(x-1)y^2 . - Michael Somos, May 23 2005
G.f. (for offset 0): (-1+x+(1-6*x+5*x^2)^(1/2))/(2*(-x+x^2)).
G.f. =z*c(z/(1-z))/(1-z) = 1/2 - (1/2)sqrt(1-4z/(1-z)), where c(z)=(1-sqrt(1-4z))/(2z) is the Catalan function (follows from Michael Somos' first comment). - Emeric Deutsch, Aug 12 2007
G.f.: 1/(1-2x-x^2/(1-3x-x^2/(1-3x-x^2/(1-3x-x^2/(1-3x-x^2/(1-.... (continued fraction). - Paul Barry, Apr 19 2009
a(n) = Sum_{k, 0<=k<=n} A091965(n,k)*(-1)^k. - Philippe Deléham, Nov 28 2009
E.g.f.: exp(3x)*(I_0(2x)-I_1(2x)), where I_k(x) is a modified Bessel function of the first kind. - Emanuele Munarini, Apr 15 2011
If we prefix sequence with an additional term a(0)=1, g.f. is (3-3*x-sqrt(1-6*x+5*x^2))/(2*(1-x)). [See Kim, 2011] - N. J. A. Sloane, May 13 2011
From Gary W. Adamson, Jul 21 2011: (Start)
a(n) = upper left term in M^(n-1), M = an infinite square production matrix as follows:
  2, 1, 0, 0, 0, 0, ...
  1, 2, 1, 0, 0, 0, ...
  1, 1, 2, 1, 0, 0, ...
  1, 1, 1, 2, 1, 0, ...
  1, 1, 1, 1, 2, 1, ...
  1, 1, 1, 1, 1, 2, ...
  ... (End)
G.f. satisfies: A(x) = Sum_{n>=0} x^n * (1 - A(x)^(n+1))/(1 - A(x)); offset=0. - Paul D. Hanna, Nov 07 2011
G.f.: 1/x - 1/x/Q(0), where Q(k)= 1 + (4*k+1)*x/((1-x)*(k+1) - x*(1-x)*(2*k+2)*(4*k+3)/(x*(8*k+6)+(2*k+3)*(1-x)/Q(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 14 2013
G.f.: (1-x - (1-5*x)*G(0))/(2*x*(1-x)), where G(k)= 1 + 4*x*(4*k+1)/( (4*k+2)*(1-x) - 2*x*(1-x)*(2*k+1)*(4*k+3)/(x*(4*k+3) + (1-x)*(k+1)/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 25 2013
Asymptotics (for offset 0): a(n) ~ 5^(n+3/2)/(8*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Jun 28 2013
G.f.: G(0)/(1-x), where G(k) = 1 + (4*k+1)*x/((k+1)*(1-x) - 2*x*(1-x)*(k+1)*(4*k+3)/(2*x*(4*k+3) + (2*k+3)*(1-x)/G(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Jan 29 2014
a(n) = JacobiP(n-1,1,-n-1/2,9)/n. - Peter Luschny, Sep 23 2014
0 = +a(n)*(+25*a(n+1) -50*a(n+2) +15*a(n+3)) +a(n+1)*(-10*a(n+1) +31*a(n+2) -14*a(n+3)) +a(n+2)*(+2*a(n+2) +a(n+3)) for all n in Z. - Michael Somos, Jan 17 2018
a(n+1) = (2/Pi) * Integral_{x = -1..1} (m + 4*x^2)^n*sqrt(1 - x^2) dx at m = 1. In general, the integral, qua sequence in n, gives the m-th binomial transform of the Catalan numbers. - Peter Bala, Jan 26 2020

A091965 Triangle read by rows: T(n,k) = number of lattice paths from (0,0) to (n,k) that do not go below the line y=0 and consist of steps U=(1,1), D=(1,-1) and three types of steps H=(1,0) (left factors of 3-Motzkin steps).

Original entry on oeis.org

1, 3, 1, 10, 6, 1, 36, 29, 9, 1, 137, 132, 57, 12, 1, 543, 590, 315, 94, 15, 1, 2219, 2628, 1629, 612, 140, 18, 1, 9285, 11732, 8127, 3605, 1050, 195, 21, 1, 39587, 52608, 39718, 19992, 6950, 1656, 259, 24, 1, 171369, 237129, 191754, 106644, 42498, 12177, 2457

Offset: 0

Emeric Deutsch, Mar 13 2004

T(n,0) = A002212(n+1), T(n,1) = A045445(n+1); row sums give A026378.
The inverse is A207815. - Gary W. Adamson, Dec 17 2006 [corrected by Philippe Deléham, Feb 22 2012]
Reversal of A084536. - Philippe Deléham, Mar 23 2007
Triangle T(n,k), 0 <= k <= n, read by rows given by T(0,0)=1, T(n,k)=0 if k < 0 or if k > n, T(n,0) = 3*T(n-1,0) + T(n-1,1), T(n,k) = T(n-1,k-1) + 3*T(n-1,k) + T(n-1,k+1) for k >= 1. - Philippe Deléham, Mar 27 2007
This triangle belongs to the family of triangles defined by T(0,0)=1, T(n,k)=0 if k < 0 or if k > n, T(n,0) = x*T(n-1,0) + T(n-1,1), T(n,k) = T(n-1,k-1) + y*T(n-1,k) + T(n-1,k+1) for k >= 1. Other triangles arise by choosing different values for (x,y): (0,0) -> A053121; (0,1) -> A089942; (0,2) -> A126093; (0,3) -> A126970; (1,0)-> A061554; (1,1) -> A064189; (1,2) -> A039599; (1,3) -> A110877; (1,4) -> A124576; (2,0) -> A126075; (2,1) -> A038622; (2,2) -> A039598; (2,3) -> A124733; (2,4) -> A124575; (3,0) -> A126953; (3,1) -> A126954; (3,2) -> A111418; (3,3) -> A091965; (3,4) -> A124574; (4,3) -> A126791; (4,4) -> A052179; (4,5) -> A126331; (5,5) -> A125906. - Philippe Deléham, Sep 25 2007
5^n = (n-th row terms) dot (first n+1 terms in (1,2,3,...)). Example for row 4: 5^4 = 625 = (137, 132, 57, 12, 1) dot (1, 2, 3, 4, 5) = (137 + 264 + 171 + 48 + 5) = 625. - Gary W. Adamson, Jun 15 2011
Riordan array ((1-3*x-sqrt(1-6*x+5*x^2))/(2*x^2), (1-3*x-sqrt(1-6*x+5*x^2))/(2*x)). - Philippe Deléham, Feb 19 2012

			Triangle begins:
     1;
     3,    1;
    10,    6,    1;
    36,   29,    9,    1;
   137,  132,   57,   12,    1;
   543,  590,  315,   94,   15,    1;
  2219, 2628, 1629,  612,  140,   18,    1;
T(3,1)=29 because we have UDU, UUD, 9 HHU paths, 9 HUH paths and 9 UHH paths.
Production matrix begins
  3, 1;
  1, 3, 1;
  0, 1, 3, 1;
  0, 0, 1, 3, 1;
  0, 0, 0, 1, 3, 1;
  0, 0, 0, 0, 1, 3, 1;
  0, 0, 0, 0, 0, 1, 3, 1;
  0, 0, 0, 0, 0, 0, 1, 3, 1;
  0, 0, 0, 0, 0, 0, 0, 1, 3, 1;
  0, 0, 0, 0, 0, 0, 0, 0, 1, 3, 1;
- _Philippe Deléham_, Nov 07 2011

A. Nkwanta, Lattice paths and RNA secondary structures, DIMACS Series in Discrete Math. and Theoretical Computer Science, 34, 1997, 137-147.

Vincenzo Librandi, Rows n = 0..100, flattened
Shu-Chiuan Chang and Robert Shrock, Structure of the Partition Function and Transfer Matrices for the Potts Model in a Magnetic Field on Lattice Strips, J. Stat. Physics 137 (2009) 667, table 5.
Helmut Prodinger, The amplitude of Motzkin paths, arXiv:2104.07596 [math.CO], 2021. Mentions this sequence.
Helmut Prodinger, Multi-edge trees and 3-coloured Motzkin paths: bijective issues, arXiv:2105.03350 [math.CO], 2021.
Helmut Prodinger, Weighted unary-binary trees, Hex-trees, marked ordered trees, and related structures, arXiv:2106.14782 [math.CO], 2021.

Cf. A002212, A045445, A026378.

Cf. A123965.

nmax = 9; t[n_, k_] := ((k+1)*n!*Hypergeometric2F1[k+3/2, k-n, 2k+3, -4]) / ((k+1)!*(n-k)!); Flatten[ Table[ t[n, k], {n, 0, nmax}, {k, 0, n}]] (* Jean-François Alcover, Nov 14 2011, after Vladimir Kruchinin *)
T[0, 0, x_, y_] := 1; T[n_, 0, x_, y_] := x*T[n - 1, 0, x, y] + T[n - 1, 1, x, y]; T[n_, k_, x_, y_] := T[n, k, x, y] = If[k < 0 || k > n, 0,
T[n - 1, k - 1, x, y] + y*T[n - 1, k, x, y] + T[n - 1, k + 1, x, y]];
Table[T[n, k, 3, 3], {n, 0, 10}, {k, 0, n}] // Flatten (* G. C. Greubel, May 22 2017 *)

T(n,k):=(k+1)*sum((binomial(2*(m+1),m-k)*binomial(n,m))/(m+1),m,k,n); /* Vladimir Kruchinin, Oct 08 2011 */

@CachedFunction
def A091965(n,k):
    if n==0 and k==0: return 1
    if k<0 or k>n: return 0
    if k==0: return 3*A091965(n-1,0)+A091965(n-1,1)
    return A091965(n-1,k-1)+3*A091965(n-1,k)+A091965(n-1,k+1)
for n in (0..7):
    [A091965(n,k) for k in (0..n)] # Peter Luschny, Nov 05 2012

G.f.: G = 2/(1 - 3*z - 2*t*z + sqrt(1-6*z+5*z^2)). Alternatively, G = M/(1 - t*z*M), where M = 1 + 3*z*M + z^2*M^2.
Sum_{k>=0} T(m, k)*T(n, k) = T(m+n, 0) = A002212(m+n+1). - Philippe Deléham, Sep 14 2005
The triangle may also be generated from M^n * [1,0,0,0,...], where M = an infinite tridiagonal matrix with 1's in the super and subdiagonals and [3,3,3,...] in the main diagonal. - Gary W. Adamson, Dec 17 2006
Sum_{k=0..n} T(n,k)*(k+1) = 5^n. - Philippe Deléham, Mar 27 2007
Sum_{k=0..n} T(n,k)*x^k = A117641(n), A033321(n), A007317(n), A002212(n+1), A026378(n+1) for x = -3, -2, -1, 0, 1 respectively. - Philippe Deléham, Nov 28 2009
T(n,k) = (k+1)*Sum_{m=k..n} binomial(2*(m+1),m-k)*binomial(n,m)/(m+1). - Vladimir Kruchinin, Oct 08 2011
The n-th row polynomial R(n,x) equals the n-th degree Taylor polynomial of the function (1 - x^2)*(1 + 3*x + x^2)^n expanded about the point x = 0. - Peter Bala, Sep 06 2022

A110877 Triangle T(n,k), 0 <= k <= n, read by rows, defined by: T(0,0) = 1, T(n,k) = 0 if n= 1: T(n,k) = T(n-1,k-1) + x*T(n-1,k) + T(n-1,k+1) with x = 3.

Original entry on oeis.org

1, 1, 1, 2, 4, 1, 6, 15, 7, 1, 21, 58, 37, 10, 1, 79, 232, 179, 68, 13, 1, 311, 954, 837, 396, 108, 16, 1, 1265, 4010, 3861, 2133, 736, 157, 19, 1, 5275, 17156, 17726, 10996, 4498, 1226, 215, 22, 1, 22431, 74469, 81330, 55212, 25716, 8391, 1893

Offset: 0

Philippe Deléham, Sep 19 2005

Similar to A064189 (x = 1) and to A039599 (x = 2).
This triangle belongs to the family of triangles defined by: T(0,0)=1, T(n,k)=0 if k < 0 or if k > n, T(n,0) = x*T(n-1,0) + T(n-1,1), T(n,k) = T(n-1,k-1) + y*T(n-1,k) + T(n-1,k+1) for k >= 1. Other triangles arise by choosing different values for (x,y): (0,0) -> A053121; (0,1) -> A089942; (0,2) -> A126093; (0,3) -> A126970; (1,0)-> A061554; (1,1) -> A064189; (1,2) -> A039599; (1,3) -> A110877; (1,4) -> A124576; (2,0) -> A126075; (2,1) -> A038622; (2,2) -> A039598; (2,3) -> A124733; (2,4) -> A124575; (3,0) -> A126953; (3,1) -> A126954; (3,2) -> A111418; (3,3) -> A091965; (3,4) -> A124574; (4,3) -> A126791; (4,4) -> A052179; (4,5) -> A126331; (5,5) -> A125906. - Philippe Deléham, Sep 25 2007
Row sums yield A126568. - Philippe Deléham, Oct 10 2007
5^n = (n-th row terms) dot (first n+1 terms in the series (1, 4, 7, 10, ...)). Example for row 4: 5^4 = 625 = (21, 58, 37, 10, 1) dot (1, 4, 7, 10, 13) = (21 + 232 + 259 + 100 + 13). - Gary W. Adamson, Jun 15 2011
Riordan array (2/(1+x+sqrt(1-6*x+5*x^2)), (1-3*x-sqrt(1-6*x+5*x^2))/(2*x)). - Philippe Deléham, Mar 04 2013

			Triangle begins:
      1;
      1,     1;
      2,     4,     1;
      6,    15,     7,     1;
     21,    58,    37,    10,     1;
     79,   232,   179,    68,    13,    1;
    311,   954,   837,   396,   108,   16,    1;
   1265,  4010,  3861,  2133,   736,  157,   19,   1;
   5275, 17156, 17726, 10996,  4498, 1226,  215,  22,  1;
  22431, 74469, 81330, 55212, 25716, 8391, 1893, 282, 25, 1;
  ...
From _Philippe Deléham_, Nov 07 2011: (Start)
Production matrix begins:
  1, 1;
  1, 3, 1;
  0, 1, 3, 1;
  0, 0, 1, 3, 1;
  0, 0, 0, 1, 3, 1;
  0, 0, 0, 0, 1, 3, 1;
  0, 0, 0, 0, 0, 1, 3, 1;
  0, 0, 0, 0, 0, 0, 1, 3, 1;
  0, 0, 0, 0, 0, 0, 0, 1, 3, 1;
  ... (End)

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

Cf. A002212, A033321, A039599, A064189.

The inverse of A126126.

A110877 := proc(n,k)
    if k > n then
        0;
    elif n= 0 then
        1;
    elif k = 0 then
        procname(n-1,0)+procname(n-1,1) ;
    else
        procname(n-1,k-1)+3*procname(n-1,k)+procname(n-1,k+1) ;
    end if;
end proc: # R. J. Mathar, Sep 06 2013

T[0, 0, x_, y_] := 1; T[n_, 0, x_, y_] := x*T[n - 1, 0, x, y] + T[n - 1, 1, x, y]; T[n_, k_, x_, y_] := T[n, k, x, y] = If[k < 0 || k > n, 0, T[n - 1, k - 1, x, y] + y*T[n - 1, k, x, y] + T[n - 1, k + 1, x, y]]; Table[T[n, k, 1, 3], {n, 0, 49}, {k, 0, n}] // Flatten (* G. C. Greubel, Apr 21 2017 *)

T(n, 0) = A033321(n) and for k >= 1: T(n, k) = Sum_{j>=1} T(n-j, k-1)*A002212(j).
Sum_{k=0..n} T(m, k)*T(n, k) = T(m+n, 0) = A033321(m+n).
The triangle may also be generated from M^n * [1,0,0,0,...], where M = an infinite tridiagonal matrix with 1's in the super and subdiagonals and (1,3,3,3,...) in the main diagonal. - Gary W. Adamson, Dec 17 2006
Sum_{k=0..n} T(n,k)*(3*k+1) = 5^n. - Philippe Deléham, Feb 26 2007
Sum_{k=0..n} T(n,k) = A126568(n). - Philippe Deléham, Oct 10 2007

A332602 Tridiagonal matrix M read by antidiagonals: main diagonal is 1,2,2,2,2,..., two adjacent diagonals are 1,1,1,1,1,...

Original entry on oeis.org

1, 1, 1, 0, 2, 0, 0, 1, 1, 0, 0, 0, 2, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0

Offset: 0

N. J. A. Sloane, Mar 06 2020, following a suggestion from Gary W. Adamson

From Gary W. Adamson, Mar 11 2020: (Start)
The upper left entry of M^n gives the Catalan numbers A000108. Extracting 2 X 2, 3 X 3, and 4 X 4 submatrices from M; then generating sequences from the upper left entries of M^n, we obtain the following sequences:
1, 1, 2, 5, 13, ... = A001519 and the convergent is 2.61803... = 2 + 2*cos(2*Pi/5) = (2*cos(Pi/5))^2.
1, 1, 2, 5, 14, 42, 131, ... = A080937 and the convergent is 3.24697... = 2 + 2*cos(2*Pi/7) = (2*cos(Pi/7))^2.
1, 1, 2, 5, 14, 42, 132, 429, 1429, ... = A080938 and the convergent is 3.53208... = 2 + 2*cos(2*Pi/9) = (2*cos(Pi/9))^2. (End)
The characteristic polynomial for the N X N main submatrix M_N is Phi(N, x) = S(N, 2-x) - S(N-1, 2-x), with Chebyshev's S polynomial (see A049310) evaluated at 2-x. Proof by determinant expansion, to obtain the recurrence Phi(N, x) - (x-2)*Phi(N-1, x) - Phi(N-2, x), for N >= 2, and Phi(0, x) = 1 and Phi(1, x) = 1 - x, that is Phi(-1, x) = 1. The trace is tr(M_N) = 1 + 2^(N-1) = A000051(N-1), and Det(M_N) = 1. - Wolfdieter Lang, Mar 13 2020
The explicit form of the characteristic polynomial for the N X N main submatrix M_N is Phi(N, x) := Det(M_N - x*1_N) = Sum_{k=0..N} binomial(N+k, 2*k)*(-x)^k = Sum_{k=0..N} A085478(N, k)*(-x)^k, for N >= 0, with Phi(0, x) := 1. Proof from the recurrence given in the preceding comment. - Wolfdieter Lang, Mar 25 2020
For the proofs of the 2 X 2, 3 X 3 and 4 X 4 conjectures, see the comments in the respective A-numbers A001519, A080937 and A080938. - Wolfdieter Lang, Mar 30 2020
Replace the main diagonal 1,2,2,2,... of the matrix M with 1,0,0,0,...; 1,1,1,1,...; 1,3,3,3,...; 1,2,1,2,...; 1,2,3,4,...; 1,0,1,0...; and 1,1,0,0,1,1,0,0,.... Take powers of M and extract the upper left terms, resulting in respectively: A001405, A001006, A033321, A176677, A006789, A090344, and A007902. - Gary W. Adamson, Apr 12 2022
The statement that the upper left entry of M^n is a Catalan number is equivalent to Exercise 41 of R. Stanley, "Catalan Numbers." - Richard Stanley, Feb 28 2023
If the upper left 1 in matrix M is replaced with 3, taking powers of the resulting matrix and extracting the upper left terms apparently results in sequence A001700. - Gary W. Adamson, Apr 03 2023

			The matrix begins:
  1, 1, 0, 0, 0, ...
  1, 2, 1, 0, 0, ...
  0, 1, 2, 1, 0, ...
  0, 0, 1, 2, 1, ...
  0, 0, 0, 1, 2, ...
  ...
The first few antidiagonals are:
  1;
  1, 1;
  0, 2, 0;
  0, 1, 1, 0;
  0, 0, 2, 0, 0;
  0, 0, 1, 1, 0, 0;
  0, 0, 0, 2, 0, 0, 0;
  0, 0, 0, 1, 1, 0, 0, 0;
  0, 0, 0, 0, 2, 0, 0, 0, 0;
  0, 0, 0, 0, 1, 1, 0, 0, 0, 0;
  ...
Characteristic polynomial of the 3 X 3 matrix M_3: Phi(3, x) = 1 - 6*x + 5*x^2 - x^3, from {A085478(3, k)}_{k=0..3} = {1, 6, 5, 1}. - _Wolfdieter Lang_, Mar 25 2020

Richard P. Stanley, "Catalan Numbers", Cambridge University Press, 2015.

Cf. A000108, A000051, A001519, A049310, A080937, A080938, A085478.
Cf. A001006, A001405, A006789, A033321, A176677, A090344, A007902.
Cf. A001333 (permanent of the matrix M).
Cf. A054142, A053123, A011973 (characteristic polynomials of submatrices of M).
Cf. A001700.

A033543 Expansion of (1 - sqrt((1-2x)(1-6x)))/(2x(2-3x)).

Original entry on oeis.org

1, 2, 5, 16, 62, 270, 1257, 6096, 30398, 154756, 800834, 4199720, 22269976, 119207942, 643277553, 3495713184, 19113486390, 105074982876, 580435709622, 3220217022144, 17935186513044, 100243540330188, 562080274898250, 3160904659483104, 17823384503589996, 100749266778698280

Offset: 0

N. J. A. Sloane

nonn

Binomial transform of A033321. - Philippe Deléham, Nov 26 2009

a(n) is the number of Motzkin paths of length n in which the (1,0)-steps at level 0 come in 2 colors and those at a higher level come in 4 colors. Example: a(3)=16 because, denoting U=(1,1), H=(1,0), and D=(1,-1), we have 2^3 = 8 paths of shape HHH, 2 paths of shape HUD, 2 paths of shape UDH, and 4 paths of shape UHD. - Emeric Deutsch, May 02 2011

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Isaac DeJager, Madeleine Naquin, Frank Seidl, Colored Motzkin Paths of Higher Order, VERUM 2019.
J. W. Layman, The Hankel Transform and Some of its Properties, J. Integer Sequences, 4 (2001), #01.1.5.

R:=PowerSeriesRing(Rationals(), 40); Coefficients(R!( (1-Sqrt((1-2*x)*(1-6*x)))/(2*x*(2-3*x)) )); // G. C. Greubel, Oct 12 2019

seq(coeff(series((1-sqrt((1-2*x)*(1-6*x)))/(2*x*(2-3*x)), x, n+2), x, n), n = 0..40); # G. C. Greubel, Oct 12 2019

CoefficientList[Series[(1-Sqrt[(1-2x)(1-6x)])/(2x(2-3x)),{x,0,40}],x] (* Harvey P. Dale, Aug 12 2012 *)

x='x+O('x^66); Vec( (1-sqrt((1-2*x)*(1-6*x)))/(2*x*(2-3*x)) ) \\ Joerg Arndt, May 04 2013

def A033543_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( (1-sqrt((1-2*x)*(1-6*x)))/(2*x*(2-3*x)) ).list()
A033543_list(40) # G. C. Greubel, Oct 12 2019

a(n) = A124575(n,0). - Philippe Deléham, Nov 26 2009
a(n) = Sum_{k=0..n} A052179(n,k)*(-2)^k. - Philippe Deléham, Nov 28 2009
From Gary W. Adamson, Jul 21 2011: (Start)
a(n) = upper left term in M^n, M = an infinite square production matrix as follows (with the main diagonal (2,3,3,3,...)):
  2, 1, 0, 0, ...
  1, 3, 1, 0, ...
  1, 1, 3, 1, ...
  1, 1, 1, 3, ...
  ... (End)
D-finite with recurrence: 2*(n+1)*a(n) = (19*n-5)*a(n-1) - 12*(4*n-5)*a(n-2) + 36*(n-2)*a(n-3). - Vaclav Kotesovec, Oct 08 2012
a(n) ~ 6^(n+1/2)/(3*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Oct 08 2012

A181229 T(n,k)=Number of length n sequences p(i=0..n-1) with 0<=p(i)<=i and having exactly k maxima.

Original entry on oeis.org

1, 0, 2, 0, 0, 6, 0, 0, 0, 21, 0, 0, 0, 3, 79, 0, 0, 0, 0, 41, 311, 0, 0, 0, 0, 0, 383, 1265, 0, 0, 0, 0, 0, 26, 3073, 5275, 0, 0, 0, 0, 0, 0, 702, 22835, 22431, 0, 0, 0, 0, 0, 0, 0, 11734, 162517, 96900, 0, 0, 0, 0, 0, 0, 0, 476, 156976, 1127399, 424068, 0, 0, 0, 0, 0, 0, 0, 0, 20956

Offset: 1

R. H. Hardin Oct 10 2010

Table starts
............1...............0.................0..................0
............2...............0.................0..................0
............6...............0.................0..................0
...........21...............3.................0..................0
...........79..............41.................0..................0
..........311.............383................26..................0
.........1265............3073...............702..................0
.........5275...........22835.............11734................476
........22431..........162517............156976..............20956
........96900.........1127399...........1847636.............541960
.......424068.........7700835..........20046784...........10783530
......1876143........52118506.........205891684..........183084580
......8377299.......350902046........2034977036.........2796467584
.....37704042......2356532130.......19567350334........39641212284
....170870106.....15813455692......184421027812.......532014841490
....779058843....106160717591.....1712803733352......6852097042820
...3571051579....713555194577....15736451922346.....85509456576162
..16447100702...4804401287757...143435108141199...1041222110654522
..76073821946..32414156949765..1299841823088167..12436416067831144
.353224531663.219174260505645.11730670902611152.146290172122139436

			All solutions for n=5 with k=2 maxima
..0..1..0..0..1....0..1..0..0..2....0..1..0..0..3....0..1..0..0..4
..0..1..0..2..0....0..1..0..2..1....0..1..0..2..2....0..1..0..2..3
..0..1..0..2..4....0..1..0..1..0....0..1..0..1..1....0..1..0..1..2
..0..1..0..1..3....0..1..0..1..4....0..1..0..3..0....0..1..0..3..1
..0..1..0..3..2....0..1..0..3..3....0..1..0..3..4....0..0..1..0..1
..0..0..1..0..2....0..0..1..0..3....0..0..1..0..4....0..0..2..0..1
..0..0..2..0..2....0..0..2..0..3....0..0..2..0..4....0..0..2..1..2
..0..0..2..1..3....0..0..2..1..4....0..1..1..0..1....0..1..1..0..2
..0..1..1..0..3....0..1..1..0..4....0..1..2..0..1....0..1..2..0..2
..0..1..2..0..3....0..1..2..0..4....0..1..2..1..2....0..1..2..1..3
..0..1..2..1..4

R. H. Hardin, Table of n, a(n) for n=1..10000

Column 1 is A033321

A092107 Triangle read by rows: T(n,k) is the number of Dyck paths of semilength n having exactly k UUU's (triple rises) where U=(1,1). Rows have 1,1,1,2,3,4,5,... entries, respectively.

Original entry on oeis.org

1, 1, 2, 4, 1, 9, 4, 1, 21, 15, 5, 1, 51, 50, 24, 6, 1, 127, 161, 98, 35, 7, 1, 323, 504, 378, 168, 48, 8, 1, 835, 1554, 1386, 750, 264, 63, 9, 1, 2188, 4740, 4920, 3132, 1335, 390, 80, 10, 1, 5798, 14355, 17028, 12507, 6237, 2200, 550, 99, 11, 1, 15511, 43252, 57816

Offset: 0

Emeric Deutsch, Mar 29 2004

Column 0 gives the Motzkin numbers (A001006), column 1 gives A014532. Row sums are the Catalan numbers (A000108).

Equal to A171380*B (without the zeros), B = A007318. - Philippe Deléham, Dec 10 2009

			T(5,2) = 5 because we have (U[UU)U]DUDDDD, (U[UU)U]DDUDDD, (U[UU)U]DDDUDD, (U[UU)U]DDDDUD and UD(U[UU)U]DDDD, where U=(1,1), D=(1,-1); the triple rises are shown between parentheses.
[1],[1],[2],[4, 1],[9, 4, 1],[21, 15, 5, 1],[51, 50, 24, 6, 1],[127, 161, 98, 35, 7, 1]
Triangle starts:
     1;
     1;
     2;
     4,    1;
     9,    4,    1;
    21,   15,    5,    1;
    51,   50,   24,    6,    1;
   127,  161,   98,   35,    7,    1;
   323,  504,  378,  168,   48,    8,    1;
   835, 1554, 1386,  750,  264,   63,    9,    1;
  2188, 4740, 4920, 3132, 1335,  390,   80,   10,    1;
  ...

Alois P. Heinz, Rows n = 0..150, flattened
Jean Luc Baril, Rigoberto Flórez, and José L. Ramirez, Generalized Narayana arrays, restricted Dyck paths, and related bijections, Univ. Bourgogne (France, 2025). See p. 14.
Michael Bukata, Ryan Kulwicki, Nicholas Lewandowski, Lara Pudwell, Jacob Roth, and Teresa Wheeland, Distributions of Statistics over Pattern-Avoiding Permutations, arXiv:1812.07112 [math.CO], 2018.
FindStat - Combinatorial Statistic Finder, The number of occurrences of the contiguous pattern [.,[.,[.,.]]].
Lara Pudwell, On the distribution of peaks (and other statistics), 16th International Conference on Permutation Patterns, Dartmouth College, 2018.
Toufik Mansour and Mark Shattuck, Counting occurrences of subword patterns in non-crossing partitions, Art Disc. Appl. Math. (2022).
A. Sapounakis, I. Tasoulas and P. Tsikouras, Counting strings in Dyck paths, Discrete Math., 307 (2007), 2909-2924.
Aristidis Sapounakis, Panagiotis Tsikouras, Ioannis Tasoulas, and Kostas Manes, Strings of Length 3 in Grand-Dyck Paths and the Chung-Feller Property, Electr. J. Combinatorics, 19 (2012), #P2.
Yidong Sun, The statistic "number of udu's" in Dyck paths, Discrete Math., 287 (2004), 177-186.

Cf. A000108, A001006, A001405, A007318, A014532, A033321, A171380, A243752, A243753.

b:= proc(x, y, t) option remember; `if`(y>x or y<0, 0,
      `if`(x=0, 1, expand(b(x-1, y-1, min(t+1,2))*
      `if`(t=2, z, 1) +b(x-1, y+1, 0))))
    end:
T:= n->(p->seq(coeff(p, z, i), i=0..degree(p)))(b(2*n, 0, 0)):
seq(T(n), n=0..12);  # Alois P. Heinz, Mar 11 2014

b[x_, y_, t_] := b[x, y, t] = If[y>x || y<0, 0, If[x == 0, 1, Expand[b[x-1, y-1, Min[t+1, 2]]*If[t == 2, z, 1] + b[x-1, y+1, 0]]]]; T[n_] := Function[{p}, Table[ Coefficient[p, z, i], {i, 0, Exponent[p, z]}]][b[2*n, 0, 0]]; Table[T[n], {n, 0, 12}] // Flatten (* Jean-François Alcover, Apr 29 2015, after Alois P. Heinz *)

G.f.: G(t, z) satisfies z(t+z-tz)G^2 - (1-z+tz)G + 1 = 0.

Sum_{k=0..n} T(n,k)*x^k = A001405(n), A001006(n), A000108(n), A033321(n) for x = -1, 0, 1, 2 respectively. - Philippe Deléham, Dec 10 2009

cp's OEIS Frontend

A171486 Riordan array (f(x), x*f(x)) where f(x) is the g.f. of A033321.

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A007317 Binomial transform of Catalan numbers.

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A091965 Triangle read by rows: T(n,k) = number of lattice paths from (0,0) to (n,k) that do not go below the line y=0 and consist of steps U=(1,1), D=(1,-1) and three types of steps H=(1,0) (left factors of 3-Motzkin steps).

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A110877 Triangle T(n,k), 0 <= k <= n, read by rows, defined by: T(0,0) = 1, T(n,k) = 0 if n= 1: T(n,k) = T(n-1,k-1) + x*T(n-1,k) + T(n-1,k+1) with x = 3.

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Maple

Mathematica

Formula

A332602 Tridiagonal matrix M read by antidiagonals: main diagonal is 1,2,2,2,2,..., two adjacent diagonals are 1,1,1,1,1,...

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A033543 Expansion of (1 - sqrt((1-2*x)*(1-6*x)))/(2*x*(2-3*x)).

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Magma

Maple

Mathematica

PARI

Sage

Formula

A181229 T(n,k)=Number of length n sequences p(i=0..n-1) with 0<=p(i)<=i and having exactly k maxima.

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A033543 Expansion of (1 - sqrt((1-2x)(1-6x)))/(2x(2-3x)).