A025238 -id:A025238 - 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.
S. J. Cyvin, J. Brunvoll, G. Xiaofeng, and Z. Fuji, Number of perifusenes with one internal vertex, Rev. Roumaine Chem., 38(1) (1993), 65-78.
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)

A045868 Expansion of g.f.: ((1 - x - sqrt(1-6x+5x^2))/(2*x))^2.

Original entry on oeis.org

1, 2, 7, 26, 101, 406, 1676, 7066, 30302, 131782, 579867, 2576982, 11550237, 52152330, 237005385, 1083211410, 4975796735, 22960105510, 106377393365, 494674698190, 2308015808015, 10801388134690, 50691017885290, 238503869991926, 1124828963516896, 5316520644648026, 25179670936870021

Offset: 0

N. J. A. Sloane

Convolution of A002212 with itself.
Number of skew Dyck paths of semilength n+1 starting with UU. 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. Example: a(2)=7 because we have UUDDUD, UUDUDD, UUDUDL, UUUDDD, UUUDDL, UUUDLD and UUUDLL. - Emeric Deutsch, May 11 2007
a(n) is also the number of path-pairs (u,v) having the following six properties: 1) the lengths of u and v sum up to 2n, 2) u and v both start at (0,0), 3) (0,0) is the only vertex that u and v have in common, 4) the steps that u can make are (1,0), (0,1) and (0,-1), 5) the steps that v can make are (1,0), (-1,0) and (0,1), 6) if A and B are the termini of u and v, respectively, then B=A+(1,-1). - Svjetlan Feretic, Jun 09 2013

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
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.
Emeric Deutsch, Emanuele Munarini, and Simone Rinaldi, Skew Dyck paths, J. Stat. Plann. Infer. 140 (8) (2010) 2191-2203.
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. 99.

Cf. A055450.

Essentially the first differences of A002212 and A025238.

[n le 2 select n else (2*(3*n-2)*Self(n-1) - 5*(n-3)*Self(n-2))/(n+1): n in [1..30]]; // G. C. Greubel, Jan 12 2024

a := n->(2/n)*sum(binomial(n,j)*binomial(2*j+1,j-1),j=1..n): 1,seq(a(n),n=1..22);

a[n_] := 2*Hypergeometric2F1[ 5/2, 1-n, 4, -4]; a[0] = 1; Table[a[n], {n, 0, 22}] (* Jean-François Alcover, Apr 30 2012, after Maple *)

a(n)=polcoeff((1-x-sqrt(1-6*x+5*x^2+x^2*O(x^n)))^2/4,n+2)

my(x='x+O('x^66)); Vec(((1-x-sqrt(1-6*x+5*x^2))/(2*x))^2) \\ Joerg Arndt, May 04 2013

def A045868(n): return 1 if n==0 else (2/n)*sum( binomial(n,j)*binomial(2*j+1,j-1) for j in range(1,n+1))
[A045868(n) for n in range(31)] # G. C. Greubel, Jan 12 2024

a(n) = (2/n)*Sum_{j=1..n} binomial(n, j)*binomial(2j+1, j-1) for n >= 1.
a(n) = A055450(n, n-1).
D-finite with recurrence: (n+2)*a(n) = (6*n+2)*a(n-1) - (5*n-10)*a(n-2). - Vladeta Jovovic, Jul 16 2004
a(n) = 2*Hypergeometric2F1(5/2, 1-n, 4, -4). - Jean-François Alcover, Apr 30 2012
a(n) ~ 2*5^(n+1/2)/(sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Oct 08 2012
G.f.: 1 - 1/x + Q(0)*(1-x)/x, 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 - 2*(1-x)/x/( G(0) + 1), where G(k) = 1 + 2*x*(4*k+1)/( (2*k+1)*(1-x) - 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 24 2013

More terms from Emeric Deutsch, May 11 2007

A126186 Triangle read by rows: T(n,k) is number of hex trees with n edges and level of first leaf (in the preorder traversal) equal to k (1 <= k <= n).

Original entry on oeis.org

3, 1, 9, 3, 6, 27, 10, 19, 27, 81, 36, 66, 90, 108, 243, 137, 245, 325, 378, 405, 729, 543, 954, 1242, 1416, 1485, 1458, 2187, 2219, 3848, 4944, 5563, 5760, 5589, 5103, 6561, 9285, 15942, 20286, 22620, 23235, 22410, 20412, 17496, 19683, 39587, 67442, 85194

Offset: 1

Emeric Deutsch, Dec 22 2006

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 middle child, or a right child (name due to an obvious bijection with certain tree-like polyhexes; see the Harary-Read reference).

			Triangle starts:
   3;
   1,   9;
   3,   6,  27;
  10,  19,  27,  81;
  36,  66,  90, 108, 243;

F. Harary and R. C. Read, The enumeration of tree-like polyhexes, Proc. Edinburgh Math. Soc. (2) 17 (1970), 1-13.

Cf. A000244, A002212, A025238, A027471, A126187.

G:=2/(2-t-3*t*z+t*sqrt(1-6*z+5*z^2))-1: Gser:=simplify(series(G,z=0,14)): for n from 1 to 10 do P[n]:=sort(coeff(Gser,z,n)) od: for n from 1 to 10 do seq(coeff(P[n],t,k),k=1..n) od; # yields sequence in triangular form

n = 10; g[t_, z_] = 2/(2 - t - 3t*z + t*Sqrt[1 - 6z + 5z^2]) - 1; Flatten[ Rest[ CoefficientList[#, t]] & /@ Rest[ CoefficientList[ Series[g[t, z], {z, 0, n}], z]]] (* Jean-François Alcover, Jul 22 2011, after g.f. *)

T(n,k) = [k/(n-k)] sum(3^(2k+2j-n)*binomial(n-k,j)*binomial(k-1+j,n-k-1-j), j=ceiling((n-2k)/2)..n-k) if 1<=k

G.f.: 2/[2-t-3tz+t*sqrt(1-6z+5z^2)]-1.

Sum of terms in row n = A002212(n+1).

T(n,1) = A025238(n); T(n,1) = A002212(n-1) for n>=2.

T(n,n) = 3^n = A000244(n); T(n,n-1) = (n-1)*3^(n-2) = A027471(n) (n>=2).

Sum_{k=1..n} k*T(n,k) = A126187(n).

A128888 Table with g.f. [1-xn-sqrt(x^2n^2-2nx+1+4x^2-4x)]/(2*x).

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 3, 0, 1, 3, 8, 10, 0, 1, 4, 15, 36, 36, 0, 1, 5, 24, 84, 176, 137, 0, 1, 6, 35, 160, 510, 912, 543, 0, 1, 7, 48, 270, 1152, 3279, 4928, 2219, 0, 1, 8, 63, 420, 2240, 8768, 21975, 27472, 9285, 0, 1, 9, 80, 616, 3936, 19605, 69504, 151905, 156864

Offset: 0

R. J. Mathar, Apr 19 2007

Column m=2 is essentially the same as A005563 or A067998 or A106230. Row n=1 is essentially the same as A025238 and A002212. The table is read along diagonals and provides the Taylor coefficient of x^m in column m. It also is the slice t=1 through the trivariate g.f. defined in A129170, which provides an implicit proof that all values are nonnegative.

			Table with rows n>=0 and columns m>=0 starts
1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...
1, 1, 3, 10, 36, 137, 543, 2219, 9285, 39587, 171369, ...
1, 2, 8, 36, 176, 912, 4928, 27472, 156864, 912832, 5394176, ...
1, 3, 15, 84, 510, 3279, 21975, 151905, 1075425, 7758777, 56839965, ...
1, 4, 24, 160, 1152, 8768, 69504, 568064, 4753920, 40537088, 350963712, ...
1, 5, 35, 270, 2240, 19605, 178535, 1675495, 16095765, 157527055, 1565170985, ...
1, 6, 48, 420, 3936, 38832, 398208, 4205904, 45459840, 500488512, 5593373184, ...
1, 7, 63, 616, 6426, 70427, 801423, 9387917, 112501809, 1372985957, 17007257421,...

Cf. A005563, A002212, A129170.

H := proc(n,x) (-x*n+1-(x^2*n^2-2*n*x+1+4*x^2-4*x)^(1/2))/(2*x) ; end: T := proc(n,m) coeftayl( H(n,x),x=0,m) ; end: for diag from 0 to 20 do for m from 0 to diag do n := diag-m ; printf("%d, ",T(n,m)) ; od ; od;

Showing 1-4 of 4 results.

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A002212 Number of restricted hexagonal polyominoes with n cells.

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A045868 Expansion of g.f.: ((1 - x - sqrt(1-6*x+5*x^2))/(2*x))^2.

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A126186 Triangle read by rows: T(n,k) is number of hex trees with n edges and level of first leaf (in the preorder traversal) equal to k (1 <= k <= n).

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A128888 Table with g.f. [1-x*n-sqrt(x^2*n^2-2*n*x+1+4*x^2-4*x)]/(2*x).

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A045868 Expansion of g.f.: ((1 - x - sqrt(1-6x+5x^2))/(2*x))^2.

A128888 Table with g.f. [1-xn-sqrt(x^2n^2-2nx+1+4x^2-4x)]/(2*x).