A104259 -id:A104259 - cp's OEIS frontend

A002212 Number of restricted hexagonal polyominoes with n cells.

1, 1, 3, 10, 36, 137, 543, 2219, 9285, 39587, 171369, 751236, 3328218, 14878455, 67030785, 304036170, 1387247580, 6363044315, 29323149825, 135700543190, 630375241380, 2938391049395, 13739779184085, 64430797069375, 302934667061301, 1427763630578197

Offset: 0

N. J. A. Sloane, Ronald C. Read

Number of Schroeder paths (i.e., consisting of steps U=(1,1), D=(1,-1), H=(2,0) and never going below the x-axis) from (0,0) to (2n,0) with no peaks at odd level. Example: a(2)=3 because we have UUDD, UHD and HH. - Emeric Deutsch, Dec 06 2003
Number of 3-Motzkin paths of length n-1 (i.e., lattice paths from (0,0) to (n-1,0) 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)). Example: a(4)=36 because we have 27 HHH paths, 3 HUD paths, 3 UHD paths and 3 UDH paths. - Emeric Deutsch, Jan 22 2004
Number of rooted, planar trees having edges weighted by strictly positive integers (multi-trees) with weight-sum n. - Roland Bacher, Feb 28 2005
Number of skew Dyck paths of semilength n. A skew Dyck path is a path in the first quadrant which begins at the origin, ends on the x-axis, consists of steps U=(1,1)(up), D=(1,-1)(down) and L=(-1,-1)(left) so that up and left steps do not overlap. The length of the path is defined to be the number of its steps. - Emeric Deutsch, May 10 2007
Equivalently, number of self-avoiding paths of semilength n in the first quadrant beginning at the origin, staying weakly above the diagonal, ending on the diagonal, and consisting of steps r=(+1,0) (right), U=(0,+1) (up), and D=(0,-1) (down). Self-avoidance implies that factors UD and DU and steps D reaching the diagonal before the end are forbidden. The a(3) = 10 such paths are UrUrUr, UrUUrD, UrUUrr, UUrrUr, UUrUrD, UUrUrr, UUUDrD, UUUrDD, UUUrrD, and UUUrrr. - Joerg Arndt, Jan 15 2024
Hankel transform of [1,3,10,36,137,543,...] is A000012 = [1,1,1,1,...]. - Philippe Deléham, Oct 24 2007
From Gary W. Adamson, May 17 2009: (Start)
Convolved with A026375, (1, 3, 11, 45, 195, ...) = A026378: (1, 4, 17, 75, ...)
(1, 3, 10, 36, 137, ...) convolved with A026375 = A026376: (1, 6, 30, 144, ...).
Starting (1, 3, 10, 36, ...) = INVERT transform of A007317: (1, 2, 5, 15, 51, ...). (End)
Binomial transform of A032357. - Philippe Deléham, Sep 17 2009
a(n) = number of rooted trees with n vertices in which each vertex has at most 2 children and in case a vertex has exactly one child, it is labeled left, middle or right. These are the hex trees of the Deutsch, Munarini, Rinaldi link. This interpretation yields the second MATHEMATICA recurrence below. - David Callan, Oct 14 2012
The left shift (1,3,10,36,...) of this sequence is the binomial transform of the left-shifted Catalan numbers (1,2,5,14,...). Example: 36 =1*14 + 3*5 + 3*2 + 1*1. - David Callan, Feb 01 2014
Number of Schroeder paths from (0,0) to (2n,0) with no level steps H=(2,0) at even level. Example: a(2)=3 because we have UUDD, UHD and UDUD. - José Luis Ramírez Ramírez, Apr 27 2015
This is the Riordan transform with the Riordan matrix A097805 (of the associated type) of the Catalan sequence A000108. See a Feb 17 2017 comment in A097805. - Wolfdieter Lang, Feb 17 2017
a(n) is the number of parking functions of size n avoiding the patterns 132 and 231. - Lara Pudwell, Apr 10 2023

			G.f. = 1 + x + 3*x^2 + 10*x^3 + 36*x^4 + 137*x^5 + 543*x^6 + 2219*x^7 + 9285*x^8 + ...

J. Brunvoll, B. N. Cyvin, and S. J. Cyvin, Studies of some chemically relevant polygonal systems: mono-q-polyhexes, ACH Models in Chem., 133 (3) (1996), 277-298, Eq 14.
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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).

T. D. Noe, Table of n, a(n) for n = 0..200
Ayomikun Adeniran and Lara Pudwell, Pattern avoidance in parking functions, Enumer. Comb. Appl. 3:3 (2023), Article S2R17.
Jean-Luc Baril, José L. Ramírez, and Lina M. Simbaqueba, Counting prefixes of skew Dyck paths, J. Int. Seq., Vol. 24 (2021), Article 21.8.2.
L. W. Beineke and R. E. Pippert, On the enumeration of planar trees of hexagons, Glasgow Math. J., 15 (1974), 131-147. See V(n).
L. W. Beineke and R. E. Pippert, On the enumeration of planar trees of hexagons, Glasgow Math. J., 15 (1974), 131-147. [Annotated scanned copy]
Johann Cigler and Christian Krattenthaler, Hankel determinants of linear combinations of moments of orthogonal polynomials, arXiv:2003.01676 [math.CO], 2020.
B. N. Cyvin et al., A class of polygonal systems representing polycyclic conjugated hydrocarbons: Catacondensed monoheptafusenes, Monat. f. Chemie, 125 (1994), 1327-1337 (see U(x)).
S. J. Cyvin, J. Brunvoll, and B. N. Cyvin, Harary-Read numbers for catafusenes: Complete classification according to symmetry, Journal of mathematical chemistry 9.1 (1992): 19-31.
S. J. Cyvin and J. Brunvoll, Generating functions for the Harary-Read numbers classified according to symmetry, Journal of mathematical chemistry 9.1 (1992): 33-38.
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.
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
Rodrigo De Castro, Andrés Ramírez, and José L. Ramírez, Applications in Enumerative Combinatorics of Infinite Weighted Automata and Graphs, arXiv preprint arXiv:1310.2449 [cs.DM], 2013. See also Sci. Annals Comp. Sci. (2014) Vol. XXIV, Issue 1, 137-171. See p. 141.
Isaac DeJager, Madeleine Naquin, and Frank Seidl, Colored Motzkin Paths of Higher Order, VERUM 2019.
Serkan Demiriz, Adem Şahin, and Sezer Erdem, Some topological and geometric properties of novel generalized Motzkin sequence spaces, Rendiconti Circ. Mat. Palermo Ser. 2 (2025) Vol. 74, No. 136. See p. 4.
Nachum Dershowitz, Touchard's Drunkard, Journal of Integer Sequences, Vol. 20 (2017), #17.1.5.
E. Deutsch, E. Munarini, and S. Rinaldi, Skew Dyck paths, J. Stat. Plann. Infer. 140 (8) (2010) 2191-2203
M. Dziemianczuk, Enumerations of plane trees with multiple edges and Raney lattice paths, Discrete Mathematics 337 (2014): 9-24.
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 p. 18.
N. S. S. Gu, N. Y. Li, and Toufik Mansour, 2-Binary trees: bijections and related issues, Discr. Math., 308 (2008), 1209-1221.
F. Harary and R. C. Read, The enumeration of tree-like polyhexes, Proc. Edinburgh Math. Soc. (2) 17 (1970), 1-13.
C. Heuberger, H. Prodinger, and S. Wagner, The height of multiple edge plane trees, arXiv preprint arXiv:1503.04749 [math.CO], 2015.
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.
C. Jean-Louis and A. Nkwanta, Some algebraic structure of the Riordan group, Linear Algebra and its Applications, Nov. 27, 2012. - _N. J. A. Sloane_, Jan 03 2013
Hana Kim and R. P. Stanley, A refined enumeration of hex trees and related polynomials, Preprint 2015.
J. W. Layman, The Hankel Transform and Some of its Properties, J. Integer Sequences, 4 (2001), #01.1.5.
Huyile Liang, Jeffrey Remmel, and Sainan Zheng, Stieltjes moment sequences of polynomials, arXiv:1710.05795 [math.CO], 2017, see pp. 12-13.
Lily L. Liu, Positivity of three-term recurrence sequences, Electronic J. Combinatorics, 17 (2010), #R57.
Rui-Li Liu and Feng-Zhen Zhao, New Sufficient Conditions for Log-Balancedness, With Applications to Combinatorial Sequences, J. Int. Seq., Vol. 21 (2018), Article 18.5.7.
Toufik Mansour and Jose Luis Ramirez, Enumration of Fuss-skew paths, Ann. Math. Inform. 55 (2022) 125-136, table 1.
Michal Medek, Möbius function of matrix posets, Bachelor Thesis, Charles Univ. (Prague, Czechia 2023).
H. D. Nguyen and D. Taggart, Mining the OEIS: Ten Experimental Conjectures, 2013; http://citeseerx.ist.psu.edu/pdf/8f2f36f22878c984775ed04368b8893879b99458. Mentions this sequence. - From _N. J. A. Sloane_, Mar 16 2014
F. Pakovich and A. K. Zvonkin, Minimum degree of the difference of two polynomials over Q, and weighted plane trees, arXiv:1306.4141 [math.NT], 2013.
F. Pakovich and A. K. Zvonkin, Minimum degree of the difference of two polynomials over Q, and weighted plane trees, Selecta Mathematica, New Ser. 2014.
J.-B. Priez, A lattice of combinatorial Hopf algebras, Application to binary trees with multiplicities, arXiv preprint arXiv:1303.5538 [math.CO], 2013.
J.-B. Priez, A lattice of combinatorial Hopf algebras, Application to binary trees with multiplicities, FPSAC 2013 Paris, France DMTCS Proc. AS, 2013, 1167-1179. See also DOI
Helmut Prodinger, Weighted unary-binary trees, Hex-trees, and Horton-Strahler numbers revisited, arXiv:2106.14782 [math.CO], 2021.
Helmut Prodinger, Multi-edge trees and 3-coloured Motzkin paths: bijective issues, arXiv:2105.03350 [math.CO], 2021.
Helmut Prodinger, Partial skew Dyck paths---a kernel method approach, arXiv:2108.09785 [math.CO], 2021.
Helmut Prodinger, Prefixes of Stanley's Catalan paths with odd returns to the x-axis -- standard version and skew Catalan-Stanley paths, arXiv:2402.01429 [math.CO], 2023.
R. C. Read, Letter to N. J. A. Sloane, Feb 12 1971 (includes 40 terms of A002212 and A002216)
E. Rowland and R. Yassawi, Automatic congruences for diagonals of rational functions, arXiv preprint arXiv:1310.8635 [math.NT], 2013.
A. Sapounakis and P. Tsikouras, On k-colored Motzkin words, Journal of Integer Sequences, Vol. 7 (2004), Article 04.2.5.
R. A. Sulanke, Moments of generalized Motzkin paths, J. Integer Sequences, Vol. 3 (2000), #00.1.
Hua Sun and Yi Wang, A Combinatorial Proof of the Log-Convexity of Catalan-Like Numbers, J. Int. Seq. 17 (2014) # 14.5.2.
Y. Sun and Z. Wang, Consecutive pattern avoidances in non-crossing trees, Graph. Combinat. 26 (2010) 815-832, table 1, {uu,ud}
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.
Y. Wang and Z.-H. Zhang, Combinatorics of Generalized Motzkin Numbers, J. Int. Seq. 18 (2015) # 15.2.4.
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.
A. K. Zvonkin, Enumeration of Weighted Plane Trees, 2013.
Index entries for reversions of series

Cf. A025238, A000108, A001006, A007317, A026376, A026375.
First differences of A007317.
Row sums of triangle A104259.
Cf. A005572, A025230, A182401.

I:= [1,3]; [1] cat [n le 2 select I[n]  else ((6*n-3)*Self(n-1)-5*(n-2)*Self(n-2)) div (n+1): n in [1..30]]; // Vincenzo Librandi, Jun 15 2015

t1 := series(1+ (1-3*x-(1-x)^(1/2)*(1-5*x)^(1/2))/(2*x), x, 50):
A002212_list := len -> seq(coeff(t1,x,n),n=0..len): A002212_list(40);
a[0] := 1: a[1] := 1: for n from 2 to 50 do a[n] := (3*(2*n-1)*a[n-1]-5*(n-2)*a[n-2])/(n+1) od: print(convert(a,list)); # Zerinvary Lajos, Jan 01 2007
a := n -> `if`(n=0,1,simplify(GegenbauerC(n-1, -n, -3/2)/n)):
seq(a(n), n=0..23); # Peter Luschny, May 09 2016

InverseSeries[Series[(y)/(1+3*y+y^2), {y, 0, 24}], x] (* then A(x)=1+y(x) *) (* Len Smiley, Apr 14 2000 *)
(* faster *)
a[0]=1;a[1]=1;
a[n_]/;n>=2 := a[n] = a[n-1] +  Sum[a[i]a[n-1-i],{i,0,n-1}];
Table[a[n],{n,0,14}] (* See COMMENTS above, [David Callan, Oct 14 2012] *)
(* fastest *)
s[0]=s[1]=1;
s[n_]/;n>=2 := s[n] = (3(2n-1)s[n-1]-5(n-2)s[n-2])/(n+1);
Table[s[n],{n,0,14 }] (* See Deutsch, Munarini, Rinaldi link, [David Callan, Oct 14 2012] *)
(* 2nd fastest *)
a[n_] := Hypergeometric2F1[3/2, 1-n, 3, -4]; a[0]=1; Table[a[n], {n, 0, 14}]  (* Jean-François Alcover, May 16 2013 *)
CoefficientList[Series[(1 - x - Sqrt[1 - 6x + 5x^2])/(2x), {x, 0, 20}], x] (* Nikolaos Pantelidis, Jan 30 2023 *)

makelist(sum(binomial(n,k)*binomial(n-k,k)*3^(n-2*k)/(k+1),k,0,n/2),n,0,24); /* for a(n+1) */ /* Emanuele Munarini, May 18 2011 */

{a(n) = polcoeff( (1 - x - sqrt(1 - 6*x + 5*x^2 + x^2 * O(x^n))) / 2, n+1)};

{a(n) = if( n<1, n==0, polcoeff( serreverse( x / (1 + 3*x + x^2) + x * O(x^n)), n))}; /* Michael Somos */

my(N=66,x='x+O('x^N)); Vec((1 - x - sqrt(1-6*x+5*x^2))/(2*x)) \\ Joerg Arndt, Jan 13 2024

def A002212():
    x, y, n = 1, 1, 1
    while True:
        yield x
        n += 1
        x, y = y, ((6*n - 3)*y - (5*n - 10)*x) / (n + 1)
a = A002212()
[next(a) for i in range(24)]  # Peter Luschny, Oct 12 2013

a(0)=1, for n > 0: a(n) = Sum_{j=0..n-1} Sum_{i=0..j} a(i)*a(j-i). G.f.: A(x) = 1 + x*A(x)^2/(1-x). - Mario Catalani (mario.catalani(AT)unito.it), Jun 19 2003
a(n) = Sum_{i=ceiling((n-1)/2)..n-1} (3^(2i+1-n)*binomial(n, i)*binomial(i, n-i-1))/n. - Emeric Deutsch, Jul 23 2002
a(n) = Sum_{k=1..n} binomial(2k, k)*binomial(n-1, k-1)/(k+1), i.e., binomial transform of the Catalan numbers 1, 2, 5, 14, 42, ... (A000108). a(n) = Sum_{k=0..floor((n-1)/2)} 3^(n-1-2*k)*binomial(2k, k)*binomial(n-1, 2k)/(k+1). - Emeric Deutsch, Aug 05 2002
D-finite with recurrence: a(1)=1, a(n) = (3(2n-1)*a(n-1)-5(n-2)*a(n-2))/(n+1) for n > 1. - Emeric Deutsch, Dec 18 2002
a(n) is asymptotic to c*5^n/n^(3/2) with c=0.63.... - Benoit Cloitre, Jun 23 2003
In closed form, c = (1/2)*sqrt(5/Pi) = 0.63078313050504... - Vaclav Kotesovec, Oct 04 2012
Reversion of Sum_{n>0} a(n)x^n = -Sum_{n>0} A001906(n)(-x)^n.
G.f. A(x) satisfies xA(x)^2 + (1-x)(1-A(x)) = 0.
G.f.: (1 - x - sqrt(1 - 6x + 5x^2))/(2x). For n > 1, a(n) = 3*a(n-1) + Sum_{k=1..n-2} a(k)*a(n-k-1). - John W. Layman, Feb 22 2001
The Hankel transform of this sequence gives A001519 = 1, 2, 5, 13, 34, 89, ... E.g., Det([1, 1, 3, 10, 36; 1, 3, 10, 36, 137; 3, 10, 36, 137, 543; 10, 36, 137, 543, 2219; 36, 137, 543, 2219, 9285 ])= 34. - Philippe Deléham, Jan 25 2004
a(m+n+1) = Sum_{k>=0} A091965(m, k)*A091965(n, k) = A091965(m+n, 0). - Philippe Deléham, Sep 14 2005
a(n+1) = Sum_{k=0..n} 2^(n-k)*M(k)*binomial(n,k), where M(k) = A001006(k) is the k-th Motzkin number (from here it follows that a(n+1) and M(n) have the same parity). - Emeric Deutsch, May 10 2007
a(n+1) = Sum_{k=0..n} A097610(n,k)*3^k. - Philippe Deléham, Oct 02 2007
G.f.: 1/(1-x/(1-x-x/(1-x/(1-x-x/(1-x/(1-x-x/(1-... (continued fraction). - Paul Barry, May 16 2009
G.f.: (1-x)/(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, Oct 17 2009
G.f.: 1/(1-z/(1-z/(1-z/(...)))) where z=x/(1-x) (continued fraction); more generally g.f. C(x/(1-x)) where C(x) is the g.f. for the Catalan numbers (A000108). - Joerg Arndt, Mar 18 2011
a(n) = -5^(1/2)/(10*(n+1)) * (5*hypergeom([1/2, n], [1], 4/5) -3*hypergeom([1/2, n+1], [1], 4/5)) (for n>0). - Mark van Hoeij, Nov 12 2009
For n >= 1, a(n) = (1/(2*Pi))*Integral_{x=1..5} x^(n-1)*sqrt((x-1)*(5-x)) dx. - Groux Roland, Mar 16 2011
a(n+1) = [x^n](1-x^2)(1+3*x+x^2)^n. - Emanuele Munarini, May 18 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 (with 3,2,2,2,... as the main diagonal):
  3, 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, 0, ...
  ...
Alternatively, let M = the previous matrix but change the 3 to a 2. Then a(n) = sum of top row terms of M^(n-1). (End)
a(n) = hypergeometric([1-n,3/2],[3],-4), for n>0. - Peter Luschny, Aug 15 2012
a(n) = GegenbauerC(n-1, -n, -3/2)/n for n >= 1. - Peter Luschny, May 09 2016
E.g.f.: 1 + Integral (exp(3*x) * BesselI(1,2*x) / x) dx. - Ilya Gutkovskiy, Jun 01 2020
G.f.: 1 + x/G(0) with G(k) = (1 - 3*x - x^2/G(k+1)) (continued fraction). - Nikolaos Pantelidis, Dec 12 2022
From Peter Bala, Feb 03 2024: (Start)
G.f.: 1 + x/(1 - x) * c(x/(1 - x))^2 = 1 + x/(1 - 5*x) * c(-x/(1 - 5*x))^2, where c(x) = (1  - sqrt(1 - 4*x))/(2*x) is the g.f. of the Catalan numbers A000108.
a(n+1) = Sum_{k = 0..n} binomial(n, k)*Catalan(k+1).
a(n+1) = hypergeom([-n, 3/2], [3], -4).
a(n+1) = 5^n * Sum_{k = 0..n} (-5)^(-k)*binomial(n, k)*Catalan(k+1).
a(n+1) = 5^n * hypergeom([-n, 3/2], [3], 4/5). (End)

A007317 Binomial transform of Catalan numbers.

Original entry on oeis.org

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

J. Brunvoll et al., Studies of some chemically relevant polygonal systems: mono-q-polyhexes, ACH Models in Chem., 133 (3) (1996), 277-298, Eq. 15.
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T. D. Noe, Table of n, a(n) for n=1..200
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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.
Paul Barry, Riordan arrays, generalized Narayana triangles, and series reversion, Linear Algebra and its Applications, 491 (2016) 343-385.
Paul Barry, On the Central Antecedents of Integer (and Other) Sequences, J. Int. Seq., Vol. 23 (2020), Article 20.8.3.
Paul Barry and A. Hennessy, The Euler-Seidel Matrix, Hankel Matrices and Moment Sequences, J. Int. Seq. 13 (2010) # 10.8.2.
Andrew M. Baxter and Lara K. Pudwell, Ascent sequences avoiding pairs of patterns, 2015.
Janusz Brzozowski and Marek Szykula, Large Aperiodic Semigroups, arXiv preprint arXiv:1401.0157 [cs.FL], 2013-2014.
David Callan, Pattern avoidance in "flattened" partitions , arXiv:0802.2275 [math.CO], 2008.
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.
Giulio Cerbai, Pattern-avoiding modified ascent sequences, arXiv:2401.10027 [math.CO], 2024. See p. 17.
Giulio Cerbai, Anders Claesson, Luca Ferrari, and Einar Steingrímsson, Sorting with pattern-avoiding stacks: the 132-machine, arXiv:2006.05692 [math.CO], 2020.
Xiang-Ke Chang, X.-B. Hu, H. Lei, and Y.-N. Yeh, Combinatorial proofs of addition formulas, The Electronic Journal of Combinatorics, 23(1) (2016), #P1.8.
Harry Crane, Left-right arrangements, set partitions, and pattern avoidance, Australasian Journal of Combinatorics, 61(1) (2015), 57-72.
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

Inverse of (1, x(1-x)/(1+x-x^2)). Row sums are A002212. Augmented version of A104259.

Triangle T(n,k) given by [0,F(3)/F(1),F(1)/F(3),F(5)/F(3),F(3)/F(5),F(7)/F(5),F(5)/F(7),...] DELTA [1,0,0,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938, F(n)=A000045(n) (Fibonacci numbers). - Philippe Deléham, Jan 31 2009

cp's OEIS Frontend

A190737 Diagonal sums of the Riordan matrix A104259.

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A190738 Central coefficients of the Riordan matrix A104259.

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A002212 Number of restricted hexagonal polyominoes with n cells.

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A007317 Binomial transform of Catalan numbers.

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A154930 Inverse of Fibonacci convolution array A154929.

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A171568 Riordan array (f(x), x*f(x)) where f(x) is the g.f. of A064613.

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

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