A005572 -id:A005572 - 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)

A052179 Triangle of numbers arising in enumeration of walks on cubic lattice.

Original entry on oeis.org

1, 4, 1, 17, 8, 1, 76, 50, 12, 1, 354, 288, 99, 16, 1, 1704, 1605, 700, 164, 20, 1, 8421, 8824, 4569, 1376, 245, 24, 1, 42508, 48286, 28476, 10318, 2380, 342, 28, 1, 218318, 264128, 172508, 72128, 20180, 3776, 455, 32, 1, 1137400, 1447338

Offset: 0

N. J. A. Sloane, Jan 26 2000

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) = 4*T(n-1,0) + T(n-1,1), T(n,k) = T(n-1,k-1) + 4*T(n-1,k) + T(n-1,k+1) for k >= 1. - Philippe Deléham, Mar 27 2007
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 four types of steps H=(1,0); example: T(3,1)=50 because we have UDU, UUD, 16 HHU paths, 16 HUH paths and 16 UHH paths. - Philippe Deléham, Sep 25 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
Riordan array ((1-4x-sqrt(1-8x+12x^2))/(2x^2), (1-4x-sqrt(1-8x+12x^2))/(2x)). Inverse of A159764. - Paul Barry, Apr 21 2009
6^n = (n-th row terms) dot (first n+1 terms in (1,2,3,...)). Example: 6^3 =  216 = (76, 50, 12, 1) dot (1, 2, 3, 4) = (76 + 100 + 36 + 4) = 216. - Gary W. Adamson, Jun 15 2011
A subset of the "family of triangles" (Deléham comment of Sep 25 2007) is the succession of binomial transforms beginning with triangle A053121, (0,0); giving -> A064189, (1,1); -> A039598, (2,2); -> A091965, (3,3); -> A052179, (4,4); -> A125906, (5,5) ->, etc.; generally the binomial transform of the triangle generated from (n,n) = that generated from ((n+1),(n+1)). - Gary W. Adamson, Aug 03 2011

			Triangle begins:
    1;
    4,   1;
   17,   8,   1;
   76,  50,  12,   1;
  354, 288,  99,  16,   1;
  ...
Production matrix begins:
  4, 1;
  1, 4, 1;
  0, 1, 4, 1;
  0, 0, 1, 4, 1;
  0, 0, 0, 1, 4, 1;
  0, 0, 0, 0, 1, 4, 1;
  0, 0, 0, 0, 0, 1, 4, 1;
- _Philippe Deléham_, Nov 04 2011

G. C. Greubel, Table of n, a(n) for the first 100 rows, flattened
Rigoberto Flórez, Leandro Junes, José L. Ramírez, Further Results on Paths in an n-Dimensional Cubic Lattice, Journal of Integer Sequences, Vol. 21 (2018), Article 18.1.2.
R. K. Guy, Catwalks, Sandsteps and Pascal Pyramids, J. Integer Seqs., Vol. 3 (2000), #00.1.6.

Cf. A039598, A053121, A064189, A091965.

T:= proc(n, k) option remember; `if`(min(n, k)<0, 0,
     `if`(max(n, k)=0, 1, T(n-1, k-1)+4*T(n-1, k)+T(n-1, k+1)))
    end:
seq(seq(T(n,k), k=0..n), n=0..10);  # Alois P. Heinz, Oct 28 2021

t[0, 0] = 1; t[n_, k_] /; k < 0 || k > n = 0; t[n_, 0] := t[n, 0] = 4*t[n-1, 0] + t[n-1, 1]; t[n_, k_] := t[n, k] = t[n-1, k-1] + 4*t[n-1, k] + t[n-1, k+1]; Flatten[ Table[t[n, k], {n, 0, 9}, {k, 0, n}]] (* Jean-François Alcover, Oct 10 2011, after Philippe Deleham *)

Sum_{k>=0} T(m, k)*T(n, k) = T(m+n, 0) = A005572(m+n). - Philippe Deléham, Sep 15 2005
n-th row = M^n * V, where M = the infinite tridiagonal matrix with all 1's in the super and subdiagonals and (4,4,4,...) in the main diagonal. E.g., Row 3 = (76, 50, 12, 1) since M^3 * V = [76, 50, 12, 1, 0, 0, 0, ...]. - Gary W. Adamson, Nov 04 2006
Sum_{k=0..n} T(n,k) = A005573(n). - Philippe Deléham, Feb 04 2007
Sum_{k=0..n} T(n,k)*(k+1) = 6^n. - Philippe Deléham, Mar 27 2007
Sum_{k=0..n} T(n,k)*x^k = A033543(n), A064613(n), A005572(n), A005573(n) for x = -2, -1, 0, 1 respectively. - Philippe Deléham, Nov 28 2009
As an infinite lower triangular matrix = the binomial transform of A091965 and 4th binomial transform of A053121. - Gary W. Adamson, Aug 03 2011
G.f.: 2/(1 - 4*x - 2*x*y + sqrt(1 - 8*x + 12*x^2)). - Daniel Checa, Aug 17 2022
G.f. for the m-th column: x^m*(A(x))^(m+1), where A(x) is the g.f. of the sequence counting the walks on the cubic lattice starting and finishing on the xy plane and never going below it (A005572). Explicitly, the g.f. is x^m*((1 - 4*x - sqrt(1 - 8*x + 12*x^2))/(2*x^2))^(m+1). - Daniel Checa, Aug 28 2022

A081671 Expansion of e.g.f. exp(4x) * I_0(2x).

Original entry on oeis.org

1, 4, 18, 88, 454, 2424, 13236, 73392, 411462, 2325976, 13233628, 75682512, 434662684, 2505229744, 14482673832, 83940771168, 487610895942, 2838118247064, 16547996212044, 96635257790352, 565107853947444, 3308820294176016, 19395905063796312, 113814537122646432

Offset: 0

Paul Barry, Mar 28 2003

Binomial transform of A026375. Second binomial transform of A000984.
Largest coefficient of (1+4x+x^2)^n. - Paul Barry, Dec 15 2003
Row sums of triangle in A124574. - Philippe Deléham, Sep 27 2007
Also number of paths from (0,0) to (n,0) using steps U=(1,1), H=(1,0) and D=(1,-1), the H steps come in 4 colors. - N-E. Fahssi, Feb 05 2008
Diagonal of rational function 1/(1 - (x^2 + 4*x*y + y^2)). - Gheorghe Coserea, Aug 01 2018

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Hacène Belbachir, Abdelghani Mehdaoui and László Szalay, Diagonal Sums in the Pascal Pyramid, II: Applications, J. Int. Seq., Vol. 22 (2019), Article 19.3.5.
Isaac DeJager, Madeleine Naquin and Frank Seidl, Colored Motzkin Paths of Higher Order, VERUM 2019.
Nachum Dershowitz, Touchard's Drunkard, Journal of Integer Sequences, Vol. 20 (2017), #17.1.5.
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.
Francesc Fite and Andrew V. Sutherland, Sato-Tate distributions of twists of y^2= x^5-x and y^2= x^6+1, arXiv preprint arXiv:1203.1476 [math.NT], 2012. - From _N. J. A. Sloane_, Sep 14 2012
Rigoberto Flórez, Leandro Junes and José L. Ramírez, Further Results on Paths in an n-Dimensional Cubic Lattice, Journal of Integer Sequences, Vol. 21 (2018), Article 18.1.2.
Tony D. Noe, On the Divisibility of Generalized Central Trinomial Coefficients, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.7.
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. 97.

Column 4 of A292627.

m-th binomial transforms of A000984: A126869 (m = -2), A002426 (m = -1 and m = -3 for signed version), A000984 (m = 0 and m = -4 for signed version), A026375 (m = 1 and m = -5 for signed version), A081671 (m = 2 and m = -6 for signed version), A098409 (m = 3 and m = -7 for signed version), A098410 (m = 4 and m = -8 for signed version), A104454 (m = 5 and m = -9 for signed version).

seq(simplify(2^n*hypergeom([-n,1/2], [1], -2)),n=0..23); # Peter Luschny, Apr 26 2016
seq(simplify(GegenbauerC(n,-n,-2)),n=0..23); # Peter Luschny, May 09 2016

Table[SeriesCoefficient[1/Sqrt[(1-2*x)*(1-6*x)],{x,0,n}],{n,0,20}] (* Vaclav Kotesovec, Oct 13 2012 *)

a(n):=coeff(expand((1+4*x+x^2)^n),x^n);
makelist(a(n),n,0,30); /* Emanuele Munarini, Apr 27 2012 */

x='x+O('x^66); Vec(1/sqrt((1-2*x)*(1-6*x))) \\ Joerg Arndt, May 07 2013

{a(n) = sum(k=0, n\2, 4^(n-2*k)*binomial(n, 2*k)*binomial(2*k, k))} \\ Seiichi Manyama, May 04 2019

a(n) = Sum_{m=0..n} Sum_{k=0..m} C(n, m)*C(m, k)*C(2k, k).
G.f.: 1/sqrt((1-2*x)*(1-6*x)). - Vladeta Jovovic, Oct 09 2003
a(n) = Sum_{k=0..n} 2^(n-k) * C(n, k) * C(2*k, k). - Paul Barry, Jan 27 2005
a(n) = Sum_{k=0..n} 6^(n-k) * (-1)^k * C(n,k) * C(2*k,k). - Paul D. Hanna, Dec 09 2018
D-finite with recurrence: n*a(n) = 4*(2*n-1)*a(n-1) - 12*(n-1)*a(n-2). - Vaclav Kotesovec, Oct 13 2012
a(n) ~ sqrt(3/(2*Pi*n))*6^n. - Vaclav Kotesovec, Oct 13 2012
a(n) = 2^n*hypergeom([-n,1/2], [1], -2). - Peter Luschny, Apr 26 2016
a(n) = GegenbauerC(n, -n, -2). - Peter Luschny, May 09 2016
a(n) = Sum_{k=0..floor(n/2)} 4^(n-2*k) * binomial(n,2*k) * binomial(2*k,k). - Seiichi Manyama, May 04 2019
a(n) = (1/Pi) * Integral_{x = -1..1} (2 + 4*x^2)^n/sqrt(1 - x^2) dx  = (1/Pi) * Integral_{x = -1..1} (6 - 4*x^2)^n/sqrt(1 - x^2) dx . - Peter Bala, Jan 27 2020
From Peter Bala, Jan 10 2022: (Start)
exp(Sum_{n >= 1} a(n)*x^n/n) = 1 + 4*x + 17*x^2 + 76*x^3 + 354*x^4 + ... is the o.g.f. of A005572.
The Gauss congruences a(n*p^k) == a(n*p^(k-1)) (mod p^k) hold for prime p and positive integers n and k. (End)
a(n) = (1/2)^n * Sum_{k=0..n} 3^k * binomial(2*k,k) * binomial(2*(n-k),n-k). - Seiichi Manyama, Aug 18 2025

A124574 Triangle read by rows: row n is the first row of the matrix M[n]^(n-1), where M[n] is the n X n tridiagonal matrix with main diagonal (3,4,4,...) and super- and subdiagonals (1,1,1,...).

Original entry on oeis.org

1, 3, 1, 10, 7, 1, 37, 39, 11, 1, 150, 204, 84, 15, 1, 654, 1050, 555, 145, 19, 1, 3012, 5409, 3415, 1154, 222, 23, 1, 14445, 28063, 20223, 8253, 2065, 315, 27, 1, 71398, 146920, 117208, 55300, 16828, 3352, 424, 31, 1, 361114, 776286, 671052, 355236, 125964, 30660, 5079, 549, 35, 1

Offset: 1

Gary W. Adamson & Roger L. Bagula, Nov 04 2006

Column 1 yields A064613. Row sums yield A081671.
Triangle T(n,k), 0 <= k <= n, defined 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) + 4*T(n-1,k) + T(n-1,k+1). - Philippe Deléham, Feb 27 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) + 4*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 from 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
6^n = ((n+1)-th row terms) dot (first n+1 odd integers). Example: 6^4 = 1296 = (150, 204, 84, 15, 1) dot (1, 3, 5, 7, 9) = (150 + 612 + 420 + 105 + 9)= 1296. - Gary W. Adamson, Jun 15 2011
From Peter Bala, Sep 06 2022: (Start)
The following assume the row and column indexing start at 0.
Riordan array (f(x), x*g(x)), where f(x) = (1 - sqrt((1 - 6*x)/(1 - 2*x)))/(2*x) is the o.g.f. of A064613 and g(x) =  (1 - 4*x - sqrt(1 - 8*x + 12*x^2))/(2*x^2) is the o.g.f. of A005572.
The n-th row polynomial R(n,x) equals the n-th degree Taylor polynomial of the function (1 - x)*(1 + 4*x + x^2)^n expanded about the point x = 0.
T(n,k) = a(n,k) - a(n,k+1), where a(n,k) = Sum_{j = 0..n} binomial(n,j)* binomial(j,n-k-j)*4^(2*j+k-n). (End)

			Row 4 is (37,39,11,1) because M[4]= [3,1,0,0;1,4,1,0;0,1,4,1;0,0,1,4] and M[4]^3=[37,39,11,1; 39, 87, 51, 12; 11, 51, 88, 50; 1, 12, 50, 76].
Triangle starts:
    1;
    3,    1
   10,    7,   1;
   37,   39,  11,   1
  150,  204,  84,  15,  1;
  654, 1050, 555, 145, 19, 1;
From _Philippe Deléham_, Nov 07 2011: (Start)
Production matrix begins:
  3, 1
  1, 4, 1
  0, 1, 4, 1
  0, 0, 1, 4, 1
  0, 0, 0, 1, 4, 1
  0, 0, 0, 0, 1, 4, 1
  0, 0, 0, 0, 0, 1, 4, 1
  0, 0, 0, 0, 0, 0, 1, 4, 1
  0, 0, 0, 0, 0, 0, 0, 1, 4, 1 (End)

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

Cf. A000108, A081671 (row sums), A124575, A124576, A052179, A064613, A005572.

with(linalg): m:=proc(i,j) if i=1 and j=1 then 3 elif i=j then 4 elif abs(i-j)=1 then 1 else 0 fi end: for n from 3 to 11 do A[n]:=matrix(n,n,m): B[n]:=multiply(seq(A[n],i=1..n-1)) od: 1; 3,1; for n from 3 to 11 do seq(B[n][1,j],j=1..n) od; # yields sequence in triangular form
T := (n,k) -> (-1)^(n-k)*simplify(GegenbauerC(n-k,-n+1,2)+GegenbauerC(n-k-1,-n+1,2 )): seq(print(seq(T(n,k),k=1..n)), n=1..10); # Peter Luschny, May 13 2016

M[n_] := SparseArray[{{1, 1} -> 3, Band[{2, 2}] -> 4, Band[{1, 2}] -> 1, Band[{2, 1}] -> 1}, {n, n}]; row[1] = {1}; row[n_] := MatrixPower[M[n], n-1] // First // Normal; Table[row[n], {n, 1, 10}] // Flatten (* Jean-François Alcover, Jan 09 2014 *)
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, 4], {n, 0, 10}, {k, 0, n}] // Flatten (* G. C. Greubel, May 22 2017 *)

Sum_{k=0..n} (-1)^(n-k)*T(n,k) = (-2)^n. - Philippe Deléham, Feb 27 2007
Sum_{k=0..n} T(n,k)*(2*k+1) = 6^n. - Philippe Deléham, Mar 27 2007
T(n,k) = (-1)^(n-k)*(GegenbauerC(n-k,-n+1,2) + GegenbauerC(n-k-1,-n+1,2)). - Peter Luschny, May 13 2016

Edited by N. J. A. Sloane, Dec 04 2006

A097610 Triangle read by rows: T(n,k) is number of Motzkin paths of length n and having k horizontal steps.

Original entry on oeis.org

1, 0, 1, 1, 0, 1, 0, 3, 0, 1, 2, 0, 6, 0, 1, 0, 10, 0, 10, 0, 1, 5, 0, 30, 0, 15, 0, 1, 0, 35, 0, 70, 0, 21, 0, 1, 14, 0, 140, 0, 140, 0, 28, 0, 1, 0, 126, 0, 420, 0, 252, 0, 36, 0, 1, 42, 0, 630, 0, 1050, 0, 420, 0, 45, 0, 1, 0, 462, 0, 2310, 0, 2310, 0, 660, 0, 55, 0, 1, 132, 0, 2772, 0

Offset: 0

Emeric Deutsch, Aug 30 2004

Row sums are the Motzkin numbers (A001006). Column 0 gives the aerated Catalan numbers (A000108).
Let P_n(x) = Sum_{k=0..n} T(n,k)*x^k. P_0(x) = 1, P_1(x) = x, P_n(x) = x*P_(n-1)(x) + Sum_{j=0..n-2} P_j(x)*P_(n-2-j)(x); essentially the same as A124027. - Philippe Deléham, Oct 03 2007
G. J. Chaitin's numbers of s-expressions of size n are given by the coefficients of polynomials p(k, x) satisfying: p(k, x) = Sum_{j=2..k-1} p(j, x)*p(k-j, x). The coefficients of these polynomials also give (essentially) the triangle shown here. - Roger L. Bagula, Oct 31 2006
Exponential Riordan array [Bessel_I(1,2x)/x,x]. - Paul Barry, Mar 24 2010
Diagonal sums are the aerated large Schroeder numbers. - Paul Barry, Apr 21 2010
Non-vanishing antidiagonals are rows of A060693. - Tom Copeland, Feb 03 2016
These polynomials are related to the Gegenbauer polynomials which in turn are specializations of the Jacobi polynomials. The o.g.f. of the Gegenbauer polynomials is 1 / [1-2tx+x^2]^a. For the generating function Gb(x,h1,h2,a) = [x / (1 + h1 x + h2 x^2)]^a, the compositional inverse in x is Gbinv(x,h1,h2,a) = [(1-h1*y) - sqrt[(1-h1*y)^2-4h2*y^2]]/(2*h2*y) with y = x^(1/a). The polynomials of this entry are generated by Gbinv(x,t,1,1). The Legendre polynomials are related to the o.g.f. Gb(x,-2t,1,1/2). Cf. A121448. - Tom Copeland, Feb 07 2016
The bivariate o.g.f. in Copeland's Jan 29 2016 formulas can be related to conformal mappings of the complex plane and a solution of the dKP hierarchy. Cf. p. 24 of the Takebe et al. paper. - Tom Copeland, May 30 2018

			Triangle begins:
1;
0,  1;
1,  0,  1;
0,  3,  0,  1;
2,  0,  6,  0,  1;
0, 10,  0, 10,  0,  1;
5,  0, 30,  0, 15,  0,  1;
Row n has n+1 terms.
T(4,2)=6 because we have HHUD, HUDH, UDHH, HUHD, UHDH, UHHD, where U=(1,1), D=(1,-1) and H=(1,0).

G. J. Chaitin, Algorithmic Information Theory, Cambridge Univ. Press, 1987, page 169.

Peter J. C. Moses, Table of n, a(n) for n = 0..1080
N. Alexeev, J. Andersen, R. Penner, and P. Zograf, Enumeration of chord diagrams on many intervals and their non-orientable analogs, arXiv:1307.0967 [math.CO], 2013-2014.
M. Artioli, G. Dattoli, S. Licciardi, and S. Pagnutti, Motzkin Numbers: an Operational Point of View, arXiv:1703.07262 [math.CO], 2017, (reverse of Table 1 on p. 3).
Marilena Barnabei, Flavio Bonetti, and Niccolò Castronuovo, Motzkin and Catalan Tunnel Polynomials, J. Int. Seq., Vol. 21 (2018), Article 18.8.8.
Paul Barry, On the inversion of Riordan arrays, arXiv:2101.06713 [math.CO], 2021.
Paul Barry, On the duals of the Fibonacci and Catalan-Fibonacci polynomials and Motzkin paths, arXiv:2101.10218 [math.CO], 2021.
Tom Copeland, The Elliptic Lie Triad: Riccati and KdV Equations, Infinigens, and Elliptic Genera
Colin Defant, Postorder Preimages, arXiv preprint arXiv:1604.01723 [math.CO], 2016.
G. Ellingsrud, Elliptic curves--Basics, course notes, Fall 2014.
P. Landweber, D. Ravenel, and R. Stong, Periodic cohomology theories defined by elliptic curves
Piera Manara and Claudio Perelli Cippo, The fine structure of 4321 avoiding involutions and 321 avoiding involutions, PU. M. A. Vol. 22 (2011), 227-238. - From _N. J. A. Sloane_, Oct 13 2012
T. Takebe, Lee-Peng Teo, and A. Zabrodin, Löwner equations and dispersionless hierarchies, arXiv:math/0605161 [math.CV], p. 24, 2006.

Cf. A001006, A000108. A124027 is an essentially identical triangle.
Cf. A011973, A049310, A055151, A060693, A180874, A121448, A125181, A126120, A134264, A145271, A238390.
Cf. A001263.

G:=(1-t*z-sqrt(1-2*t*z+t^2*z^2-4*z^2))/2/z^2:
Gser:=simplify(series(G,z=0,16)): P[0]:=1:
for n from 1 to 13 do P[n]:=sort(coeff(Gser,z^n)) od:
seq(seq(coeff(t*P[n],t^k),k=1..n+1),n=0..13);
# Maple program for the triangular array:
T:=proc(n,k) if n-k mod 2 = 0 and k<=n then n!/k!/((n-k)/2)!/((n-k)/2+1)! else 0 fi end: TT:=(n,k)->T(n-1,k-1): matrix(10,10,TT);

T[n_,k_]:=If[n>=k&&EvenQ[n-k],n!/(k!((n-k)/2)!((n-k)/2+1)!),0];
Flatten[Table[T[n,k],{n,0,20},{k,0,n}]] (* Peter J. C. Moses, Apr 06 2013 *)
T[n_,k_] := If[OddQ[n - k], 0, Binomial[n, k] CatalanNumber[(n - k)/2]]; (* Peter Luschny, Jun 06 2018 *)

G.f.: [1-tz-sqrt(1-2tz+t^2*z^2-4z^2)]/(2z^2).
T(n, k) = n!/[k!((n-k)/2)!((n-k)/2-1)! ] = A055151(n, (n-k)/2) if n-k is a nonnegative even number; otherwise T(n, k) = 0.
T(n, k) = C(n, k)*C((n-k)/2)*(1+(-1)^(n-k))/2 if k <= n, 0 otherwise. - Paul Barry, May 18 2005
T(n,k) = A121448(n,k)/2^k. - Philippe Deléham, Aug 17 2006
Sum_{k=0..n} T(n,k)*2^k = A000108(n+1). - Philippe Deléham, Aug 22 2006
Sum_{k=0..n} T(n,k)*3^k = A002212(n+1). - Philippe Deléham, Oct 03 2007
G.f.: 1/(1-x*y-x^2/(1-x*y-x^2/(1-x*y-x^2/.... (continued fraction). - Paul Barry, Dec 15 2008
Sum_{k=0..n} T(n,k)*4^k = A005572(n). - Philippe Deléham, Dec 03 2009
T(n,k) = A007318(n,k)*A126120(n-k). - Philippe Deléham, Dec 12 2009
From Tom Copeland, Jan 23 2016: (Start)
E.g.f.: M(x,t) = e^(xt) AC(t) = e^(xt) I_1(2t)/t = e(xt) * e.g.f.(A126120(t)) = e^(xt) Sum_{n>=0} t^(2n)/(n!(n+1)!) = exp[t P(.,x)].
The e.g.f. of this Appell sequence of polynomials P(n,x) is e^(xt) times the e.g.f. AC(t) of the aerated Catalan numbers A126120. AC(t) = I_1(2t)/t, where I_n(x) = T_n(d/dx) I_0(x) are the modified Bessel functions of the first kind and T_n, the Chebyshev polynomials of the first kind.
P(n,x) has the lowering and raising operators L = d/dx = D and R = d/dD log{M(x,D)} = x + d/dD log{AC(D)} = x + Sum_{n>=0} c(n) D^(2n+1)/(2n+1)! with c(n) = (-1)^n A180874(n+1), i.e., L P(n,x) = n P(n-1,x) and R P(n,x) = P(n+1,x).
(P(.,x) + y)^n = P(n,x+y) = Sum_{k=0..n} binomial(n,k) P(k,x) y^(n-k) = (b.+x+y)^n, where (b.)^k = b_k = A126120(k).
Exp(b.D) e^(xt) = exp[(x+b.)t] = exp[P(.,x)t] = e^(b.t) e^(xt) = e^(xt) AC(t).
See p. 12 of the Alexeev et al. link and A055151 for a refinement.
Shifted o.g.f: G(x,t) = [1-tx-sqrt[(1-tx)^2-4x^2]] / 2x = x + t x^2 + (1+t) x^3 + ... has the compositional inverse Ginv(x,t) = x / [1 + tx + x^2] = x - t x^2 +(-1+t^2) x^3 + (2t-t^3) x^4 + (1-3t^2+t^4) x^5 + ..., a shifted o.g.f. for the signed Chebyshev polynomials of the second kind of A049310 (cf. also the Fibonacci polynomials of A011973). Then the inversion formula of A134264, involving non-crossing partitions and free probability with their multitude of interpretations (cf. A125181 also), can be used with h_0 = 1, h_1 = t, and h_2 = 1 to interpret the coefficients of the Motzkin polynomials combinatorially.
(End)
From Tom Copeland, Jan 29 2016: (Start)
Provides coefficients of the inverse of f(x) = x / [1 + h1 x + h2 x^2], a bivariate generating function of A049310 (mod signs).
finv(x) = [(1-h1*x) - sqrt[(1-h1*x)^2-4h2*x^2]]/(2*h2*x) = x + h1 x^2 + (h2 + h1^2) x^3 + (3 h1 h2 + h1^3) x^4 + ... is a bivariate o.g.f. for this entry.
The infinitesimal generator for finv(x) is g(x) d/dx with g(x) = 1 /[df(x)/dx] = x^2 / [(f(x))^2 (1 - h2 x^2)] = (1 + h1 x + h2 x^2)^2 / (1 - h2 x^2) so that [g(x)d/dx]^n/n! x evaluated at x = 0 gives the row polynomials FI(n,h1,h2) of the compositional inverse of f(x), i.e., exp[x g(u)d/du] u |_(u=0) = finv(x) = 1 / [1 -x FI(.,h1,h2)]. Cf. A145271. E.g.,
FI(0,h1,h2) = 0
FI(1,h1,h2) = 1
FI(2,h1,h2) = 1 h1
FI(3,h1,h2) = 1 h2 + 1 h1^2
FI(4,h1,h2) = 3 h2 h1 + 1 h1^3
FI(5,h1,h2) = 2 h2^2 + 6 h2 h1^2 + 1 h1^4
FI(6,h1,h2) = 10 h2^2 h1 + 10 h2 h1^3 + 1 h1^5.
And with D = d/dh1, FI(n+1, h1,h2) = MT(n,h1,h2) = (b.y + h1)^n = Sum_{k=0..n} binomial(n,k) b(k) y^k h1^(n-k) = exp[(b.y D] (h1)^n = AC(y D) (h1)^n, where b(k) = A126120(k), y = sqrt(h2), and AC(t) is defined in my Jan 23 formulas above. Equivalently, AC(y D) e^(x h1) = exp[x MT(.,h1,h2)].
The MT polynomials comprise an Appell sequence in h1 with e.g.f. e^(h1*x) AC(xy) = exp[x MT(.,h1,h2)] with lowering operator L = d/dh1 = D, i.e., L MT(n,h1,h2) = dMT(n,h1,h2)/dh1 = n MT(n-1,h1,h2) and raising operator R = h1 + dlog{AC(y L)}/dL = h1 + Sum_{n>=0} c(n) h2^(n+1) D^(2n+1)/(2n+1)! = h1 + h2 d/dh1 - h2^2 (d/dh1)^3/3! + 5 h2^3 (d/dh1)^5/5! - ... with c(n) = (-1)^n A180874(n+1) (consistent with the raising operator in the Jan 23 formulas).
The compositional inverse finv(x) is also obtained from the non-crossing partitions of A134264 (or A125181) with h_0 = 1, h_1 = h1, h_2 = h2, and h_n = 0 for all other n.
See A238390 for the umbral compositional inverse in h1 of MT(n,h1,h2) and inverse matrix.
(End)
From Tom Copeland, Feb 13 2016: (Start)
z1(x,h1,h2) = finv(x), the bivariate o.g.f. above for this entry, is the zero that vanishes for x=0 for the quadratic polynomial Q(z;z1(x,h1,h2),z2(x,h1,h2)) = (z-z1)(z-z2) = z^2 - (z1+z2) z + (z1*z2) = z^2 - e1 z + e2 = z^2 - [(1-h1*x)/(h2*x)] z + 1/h2, where e1 and e2 are the elementary symmetric polynomials for two indeterminates.
The other zero is given by z2(x,h1,h2) = (1 - h1*x)/(h2*x) - z1(x,h1,h2) = [1 - h1*x + sqrt[(1-h1*x)^2 - 4 h2*x^2]] / (2h2*x).
The two are the nontrivial zeros of the elliptic curve in Legendre normal form y^2 = z (z-z1)(z-z2), (see Landweber et al., p. 14, Ellingsrud, and A121448), and the zeros for the Riccati equation z' = (z - z1)(z - z2), associated to soliton solutions of the KdV equation (see Copeland link).
(End)
Comparing the shifted o.g.f. S(x) = x / (1 - h_1 x + h_2 x^2) for the bivariate Chebyshev polynomials S_n(h_1,h_2) of A049310 with the shifted o.g.f. H(x) = x / ((1 - a x)(1 - b x)) for the complete homogeneous symmetric polynomials H_n(a,b) = (a^(n+1)-b^(n+1)) / (a - b) shows that S_n(h_1,h_2) = H_n(a,b) for h_1 = a + b and h_2 = ab and, conversely, a = (h_1 + sqrt(h_1^2 - 4 h_2)) / 2 and b = (h_1 - sqrt(h_1^2 - 4 h_2)) / 2. The compositional inverse about the origin of S(x) gives a bivariate o.g.f. for signed Motzkin polynomials M_n(h_1,h_2) of this entry, and that of H(x) gives one for signed Narayana polynomials N_n(a,b) of A001263, thereby relating the bivariate Motzkin and Narayana polynomials by the indeterminate transformations. E.g., M_2(h_1,h_2) = h_2 + h_1^2 = ab + (a + b)^2 = a^2 + 3 ab + b^2 = N_2(a,b). - Tom Copeland, Jan 27 2024

A025230 a(n) = a(1)a(n-1) + a(2)a(n-2) + ...+ a(n-1)*a(1) for n >= 3, with initial terms 3,1.

Original entry on oeis.org

3, 1, 6, 37, 234, 1514, 9996, 67181, 458562, 3172478, 22206420, 157027938, 1120292388, 8055001716, 58314533400, 424740506109, 3110401363122, 22888001498102, 169155516667524, 1255072594261142, 9345400450314924, 69812926066668044, 523072984217339304

Offset: 1

Clark Kimberling

Vincenzo Librandi, Table of n, a(n) for n = 1..200
Rigoberto Flórez, Leandro Junes, and José L. Ramírez, Further Results on Paths in an n-Dimensional Cubic Lattice, Journal of Integer Sequences, Vol. 21 (2018), Article 18.1.2.

For Sum_{k = 0..n} m^(n-k)*binomial(n, k)*Catalan(k+1) see A126120 (m = -2), A001006 (m = -1), A000108 (m = 0), A002212 (m = 1), A005572 (m = 2), A182401 (m = 3), A025230 (m = 4).

h := n -> simplify(4^n*hypergeom([3/2, -n], [3], -1)):
a := n -> `if`(n=1, 3, h(n-2)):
seq(a(n), n=1..21); # Peter Luschny, Feb 03 2015

Rest[CoefficientList[Series[(1-Sqrt[1-12x+32x^2])/2,{x,0,30}],x]]  (* Harvey P. Dale, Feb 22 2011 *)

a(n)=polcoeff((1-sqrt(1-12*x+32*x^2+x*O(x^n)))/2,n)

{a(n)=if(n<2, 3*(n==1), n--; polcoeff( serreverse( x/(1+6*x+x^2) +x*O(x^n) ), n))} /* Michael Somos, Oct 14 2006 */

G.f.: (1-sqrt(1-12*x+32*x^2))/2. - Michael Somos, Jun 08 2000
D-finite with recurrence n*a(n) = (12*n-18)*a(n-1) - 32*(n-3)*a(n-2) - Richard Choulet, Dec 17 2009
a(n) ~ 2^(3*n-5/2)/(sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Oct 11 2013
a(n) = 4^(n-2)*hypergeom([3/2, -n+2], [3], -1) for n>1. - Peter Luschny, Feb 03 2015
a(n+1) = GegenbauerC(n-1, -n, -3)/n for n>=1. - Peter Luschny, May 09 2016
From Peter Bala, Feb 03 2024: (Start)
G.f.: 3*x + x^2/(1 - 4*x) * c(x/(1 - 4*x))^2, where c(x) = (1 - sqrt(1 - 4*x))/(2*x) is the g.f. of the Catalan numbers A000108.
a(n+2) = Sum_{k = 0..n} 4^(n-k)*binomial(n, k)*Catalan(k+1).
G.f.: 3*x + x^2/(1 - 8*x) * c(-x/(1 - 8*x))^2.
a(n+2) = 8^n * Sum_{k = 0..n} (-8)^(-k)*binomial(n, k)*Catalan(k+1).
a(n+2) = 8^n * hypergeom([-n, 3/2], [3], 1/2).
a(n) is odd iff n is a power of 2. (End)

Name clarified by Robert C. Lyons, Feb 06 2025

A005573 Number of walks on cubic lattice (starting from origin and not going below xy plane).

Original entry on oeis.org

1, 5, 26, 139, 758, 4194, 23460, 132339, 751526, 4290838, 24607628, 141648830, 817952188, 4736107172, 27487711752, 159864676803, 931448227590, 5435879858958, 31769632683132, 185918669183370, 1089302293140564

Offset: 0

N. J. A. Sloane

Binomial transform of A026378, second binomial transform of A001700. - Philippe Deléham, Jan 28 2007

The Hankel transform of [1,1,5,26,139,758,...] is [1,4,15,56,209,...](see A001353). - Philippe Deléham, Apr 13 2007

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Isaac DeJager, Madeleine Naquin, and Frank Seidl, Colored Motzkin Paths of Higher Order, VERUM 2019.
Emeric Deutsch and Jim Brawner, Problem 10795: Three-Dimensional Lattice Walks in the Upper Half-Space, Amer. Math. Monthly, 108 (Dec. 2001), 980.
Rigoberto Flórez, Leandro Junes, and José L. Ramírez, Further Results on Paths in an n-Dimensional Cubic Lattice, Journal of Integer Sequences, Vol. 21 (2018), Article 18.1.2.
R. K. Guy, Letter to N. J. A. Sloane, May 1990
R. K. Guy, Catwalks, Sandsteps and Pascal Pyramids, J. Integer Seqs., Vol. 3 (2000), #00.1.6.
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.
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. 97.

R:=PowerSeriesRing(Rationals(), 30); Coefficients(R!( (Sqrt((1-2*x)/(1-6*x)) -1)/(2*x) )); // G. C. Greubel, May 02 2019

CoefficientList[Series[(Sqrt[(1-2x)/(1-6x)]-1)/(2x),{x,0,20}],x] (* Harvey P. Dale, Jun 24 2011 *)
a[n_] := 6^n Hypergeometric2F1[1/2, -n, 2, 2/3]; Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Apr 11 2017 *)

my(x='x+O('x^30)); Vec((sqrt((1-2*x)/(1-6*x)) -1)/(2*x)) \\ G. C. Greubel, May 02 2019

((sqrt((1-2*x)/(1-6*x)) -1)/(2*x)).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, May 02 2019

From Emeric Deutsch, Jan 09 2003; corrected by Roland Bacher: (Start)
a(n) = Sum_{i=0..n} (-1)^i*6^(n-i)*binomial(n, i)*binomial(2*i, i)/(i+1);
g.f. A(x) satisfies: x(1-6x)A^2 + (1-6x)A - 1 = 0. (End)
From Henry Bottomley, Aug 23 2001: (Start)
a(n) = 6*a(n-1) - A005572(n-1).
a(n) = Sum_{j=0..n} 4^(n-j)*binomial(n, floor(n/2))*binomial(n, j). (End)
a(n) = Sum_{k=0..n} binomial(n, k)*binomial(2*k+1, k)*2^(n-k).
a(n) = Sum_{k=0..n} (-1)^k*binomial(n, k)*Catalan(k)*6^(n-k).
D-finite with recurrence (n+1)*a(n) = (8*n+2)*a(n-1)-(12*n-12)*a(n-2). - Vladeta Jovovic, Jul 16 2004
a(n) = Sum_{k=0..n} A052179(n,k). - Philippe Deléham, Jan 28 2007
Conjecture: a(n)= 6^n * hypergeom([1/2,-n],[2], 2/3). - Benjamin Phillabaum, Feb 20 2011
From Paul Barry, Apr 21 2009: (Start)
G.f.: (sqrt((1-2*x)/(1-6*x)) - 1)/(2*x).
G.f.: 1/(1-5*x-x^2/(1-4*x-x^2/(1-4*x-x^2/(1-4*x-x^2/(1-... (continued fraction). (End)
G.f.: 1/(1 - 4*x - x*(1 - 2*x)/(1 - 2*x - x*(1 - 2*x)/(1 - 2*x - x*(1 - 2*x)/(1 - 2*x - x*(1 - 2*x)/(1...(continued fraction). - Aoife Hennessy (aoife.hennessy(AT)gmail.com), Jul 02 2010
a(n) ~ 6^(n+1/2)/sqrt(Pi*n). - Vaclav Kotesovec, Oct 05 2012
G.f.: G(0)/(2*x) - 1/(2*x), where G(k)= 1 + 4*x*(4*k+1)/( (4*k+2)*(1-2*x) - 2*x*(1-2*x)*(2*k+1)*(4*k+3)/(x*(4*k+3) + (1-2*x)*(k+1)/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 24 2013
a(n) = 2^n*hypergeom([-n, 3/2], [2], -2). - Peter Luschny, Apr 26 2016
E.g.f.: exp(4*x)*(BesselI(0,2*x) + BesselI(1,2*x)). - Ilya Gutkovskiy, Sep 20 2017

More terms from Henry Bottomley, Aug 23 2001

A182401 Number of paths from (0,0) to (n,0), never going below the x-axis, using steps U=(1,1), H=(1,0) and D=(1,-1), where the H steps come in five colors.

Original entry on oeis.org

1, 5, 26, 140, 777, 4425, 25755, 152675, 919139, 5606255, 34578292, 215322310, 1351978807, 8550394455, 54419811354, 348309105300, 2240486766555, 14476490777175, 93914850905862, 611489638708140, 3994697746533171, 26175407271617955, 171991872078871311

Offset: 0

Emanuele Munarini, Apr 27 2012

Number of 3-colored Schroeder paths from (0,0) to (2n+2,0) with no level steps H=(2,0) at even level. H-steps at odd levels are colored with one of the three colors. Example: a(2)=5 because we have UUDD, UHD (3 choices) and UDUD. - José Luis Ramírez Ramírez, Apr 27 2015

			seq(3^n * simplify(hypergeom([3/2, -n], [3], -4/3)), n = 0..20); # _Peter Bala_, Feb 04 2024

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

Cf. A001006, A002212, A005572, A025230, A126120, A257290.

CoefficientList[Series[(1-5*x-Sqrt[1-10*x+21*x^2])/(2*x^2), {x, 0, 20}], x] (* Vaclav Kotesovec, Oct 20 2012 *)
a[n_] := 5^n*Hypergeometric2F1[(1-n)/2, -n/2, 2, 4/25]; Table[a[n], {n, 0, 22}] (* Jean-François Alcover, Feb 22 2013, after 2nd formula *)

a(n):=coeff(expand((1+5*x+x^2)^(n+1)),x^n)/(n+1);
makelist(a(n),n,0,30);

x='x+O('x^66); Vec((1-5*x-sqrt(1-10*x+21*x^2))/(2*x^2)) \\ Joerg Arndt, Jun 02 2013

a(n) = [x^n] (1+5*x+x^2)^(n+1)/(n+1).
a(n) = Sum_{k=0..floor(n/2)} (binomial(n,2*k)*binomial(2*k,k)/(k+1))*5^(n-2*k).
G.f.: (1-5*x-sqrt(1-10*x+21*x^2))/(2*x^2).
Conjecture: (n+2)*a(n) +5*(-2*n-1)*a(n-1) +21*(n-1)*a(n-2)=0. - R. J. Mathar, Jul 24 2012
a(n) ~ 7^(n+3/2)/(2*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Oct 20 2012
a(n) = A125906(n,0). - Philippe Deléham, Mar 04 2013
G.f.: 1/(1 - 5*x - x^2/(1 - 5*x - x^2/(1 - 5*x - x^2/(1 - 5*x - x^2/(1 - ...))))), a continued fraction. - Ilya Gutkovskiy, Sep 21 2017
From Seiichi Manyama, Jan 15 2024: (Start)
G.f.: (1/x) * Series_Reversion( x / (1+5*x+x^2) ).
a(n) = (1/(n+1)) * Sum_{k=0..n} 3^(n-k) * binomial(n+1,n-k) * binomial(2*k+2,k). (End)
From Peter Bala, Feb 03 2024: (Start)
G.f: 1/(1 - 3*x)*c(x/(1 - 3*x))^2 = 1/(1 - 7*x)*c(-x/(1 - 7*x))^2, where c(x) = (1 - sqrt(1 - 4*x))/(2*x) is the g.f. of the Catalan numbers A000108.
a(n) = Sum_{k = 0..n} 3^(n-k)*binomial(n, k)*Catalan(k+1).
a(n) = 3^n * hypergeom([3/2, -n], [3], -4/3).
a(n) = 7^n * Sum_{k = 0..n} (-7)^(-k)*binomial(n, k)*Catalan(k+1).
a(n) = 7^n * hypergeom([3/2, -n], [3], 4/7). (End)

A025228 a(n) = a(1)a(n-1) + a(2)a(n-2) + ...+ a(n-1)*a(1) for n >= 3, with initial terms 2,1.

Original entry on oeis.org

2, 1, 4, 17, 76, 354, 1704, 8421, 42508, 218318, 1137400, 5996938, 31940792, 171605956, 928931280, 5061593709, 27739833228, 152809506582, 845646470616, 4699126915422, 26209721959656, 146681521121244, 823429928805936, 4635568494271458

Offset: 1

Clark Kimberling

nonn

Essentially A005572 shifted right twice, and 2 prepended.

Vincenzo Librandi, Table of n, a(n) for n = 1..200

Rest[CoefficientList[Series[(1-Sqrt[1-8x+12x^2])/2,{x,0,30}],x]]  (* Harvey P. Dale, Apr 20 2011 *)

a(n)=polcoeff((1-sqrt(1-8*x+12*x^2+x*O(x^n)))/2,n)

G.f.: (1-sqrt(1-8*x+12*x^2))/2. - Michael Somos, Jun 08 2000
n*a(n) = (8*n-12)*a(n-1) - 12*(n-3)*a(n-2). [Richard Choulet, Dec 16 2009]
a(n) ~ 6^(n-1/2)/(2*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Oct 11 2013

Name clarified by Robert C. Lyons, Feb 06 2025

A129400 Number of walks of length n on one 60-degree wedge of the equilateral triangular lattice. The walk can go along the walls of the wedge, but cannot cross the walls.

Original entry on oeis.org

1, 2, 8, 32, 144, 672, 3264, 16256, 82688, 427520, 2240512, 11874304, 63533056, 342712320, 1861779456, 10176823296, 55932813312, 308907737088, 1713473323008, 9541666209792, 53322206674944, 298943898451968

Offset: 0

Rebecca Xiaoxi Nie (rebecca.nie(AT)utoronto.ca), May 28 2007

Counts colored Motzkin paths where each of the steps has two possible colors. Series reversion of x/(1+2x+4x^2). - Paul Barry, Sep 04 2007

Hankel transform is 4^C(n+1,2). - Paul Barry, Oct 01 2009

			a(1) = 2 because we can go east or northeast.

Vincenzo Librandi, Table of n, a(n) for n = 0..300
Alin Bostan, Computer Algebra for Lattice Path Combinatorics, Séminaire de Combinatoire Ph. Flajolet, March 28 2013.
Alin Bostan, Calcul Formel pour la Combinatoire des Marches [The text is in English], Habilitation à Diriger des Recherches, Laboratoire d'Informatique de Paris Nord, Université Paris 13, December 2017.
Alin Bostan, Frédéric Chyzak, Mark van Hoeij, Manuel Kauers, and Lucien Pech, Hypergeometric expressions for generating functions of walks with small steps in the quarter plane. Eur. J. Comb. 61, 242-275 (2017)
Alin Bostan, Andrew Elvey Price, Anthony John Guttmann, and Jean-Marie Maillard, Stieltjes moment sequences for pattern-avoiding permutations, arXiv:2001.00393 [math.CO], 2020.

Cf. A000108, A005572.

countwalk2 := proc (i::integer, j::integer, n::integer) option remember: if n < 0 or j < 0 or i < j then 0 elif n = 0 and i = 0 and j = 0 then 1 elif n = 0 then 0 else procname(i-2, j, n-1)+procname(i+2, j, n-1)+procname(i-1, j+1, n-1)+procname(i+1, j+1, n-1)+procname(i+1, j-1, n-1)+procname(i-1, j-1, n-1) end if end proc: counter2 := proc (n::nonnegint) option remember: add(add(countwalk2(i, j, n), i = 0 .. 2*n), j = 0 .. n) end proc:
g := n -> simplify(2^n*GegenbauerC(n, -n-1, -1/2)/(n+1)):
seq(g(n), n=0..21); # Peter Luschny, May 09 2016
T := proc(n, k) option remember;
if n < 0 or k < 0 then 0
elif n = 0 then binomial(2*k, k)/(k+1)
else 2*(T(n-1, k+1) - T(n-1, k)) fi end:
a := n -> T(n, 1): seq(a(n), n=0..21); # Peter Luschny, Aug 23 2017

CoefficientList[Series[1/(8*x^2)*(1-2*x-Sqrt[1-4*x-12*x^2]), {x, 0, 20}], x] (* Vaclav Kotesovec, Oct 20 2012 *)

a(n) = 2^n*A001006(n) = Sum_{k=0..floor(n/2)} C(n,2k)*C(k)*2^(n-2k)*2^k*2^k where C(n) = A000108(n). - Paul Barry, Sep 04 2007
G.f.: 1/(1-2x-4x^2/(1-2x-4x^2/(1-2x-4x^2/(1-2x-4x^2/(1-.... (continued fraction). - Paul Barry, Oct 01 2009
G.f.: (1/(8*x^2)) * (1-2*x-(1-4*x-12*x^2)^(1/2)). - Mark van Hoeij, Nov 02 2009
E.g.f.: a(n) = n! * [x^n] exp(2*x)*BesselI(1,4*x)/(2*x). - Peter Luschny, Aug 25 2012
Recurrence: (n+2)*a(n) = 2*(2*n+1)*a(n-1) + 12*(n-1)*a(n-2) . - Vaclav Kotesovec, Oct 20 2012
a(n) ~ 3*sqrt(3)*6^n/(2*sqrt(Pi)*n^(3/2)) . - Vaclav Kotesovec, Oct 20 2012
a(n) = 2^n*GegenbauerC(n, -n-1, -1/2)/(n+1). - Peter Luschny, May 09 2016
G.f.: A(x) = 1/(1 + 2*x)*c(2*x/(1 + 2*x))^2, where c(x) = (1 - sqrt(1 - 4*x))/(2*x) is the g.f. of the Catalan numbers A000108. Cf. A005572. - Peter Bala, Aug 18 2021

cp's OEIS Frontend

A002212 Number of restricted hexagonal polyominoes with n cells.

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A052179 Triangle of numbers arising in enumeration of walks on cubic lattice.

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A081671 Expansion of e.g.f. exp(4x) * I_0(2x).

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A124574 Triangle read by rows: row n is the first row of the matrix M[n]^(n-1), where M[n] is the n X n tridiagonal matrix with main diagonal (3,4,4,...) and super- and subdiagonals (1,1,1,...).

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A097610 Triangle read by rows: T(n,k) is number of Motzkin paths of length n and having k horizontal steps.

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A025230 a(n) = a(1)*a(n-1) + a(2)*a(n-2) + ...+ a(n-1)*a(1) for n >= 3, with initial terms 3,1.

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A025230 a(n) = a(1)a(n-1) + a(2)a(n-2) + ...+ a(n-1)*a(1) for n >= 3, with initial terms 3,1.

A025228 a(n) = a(1)a(n-1) + a(2)a(n-2) + ...+ a(n-1)*a(1) for n >= 3, with initial terms 2,1.