A002057 -id:A002057 - cp's OEIS frontend

A045445 Number of nonisomorphic systems of catafusenes for the unsymmetrical schemes (group C_s) with two appendages (see references for precise definition).

0, 1, 6, 29, 132, 590, 2628, 11732, 52608, 237129, 1074510, 4893801, 22395420, 102943815, 475139070, 2201301575, 10234016880, 47731093715, 223273611810, 1047265325255, 4924606035900, 23211459517120, 109642275853176, 518959629394294, 2460993383491632, 11691102386417575

Offset: 1

N. J. A. Sloane

Number of 3-Motzkin paths of length n (i.e., lattice paths from (0,0) to (n,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)) that start with a U step. Example: a(4) = 29 because we have UDUD, UUDD, 9 UDHH paths, 9 UHDH paths and 9 UHHD paths. - Emeric Deutsch, Mar 26 2004

Here, n is the total number of hexagons in the system, which is usually denoted by h in most of the references below. In Cyvin, Brunvoll, and Cyvin (1992), Table 1, p. 28, it seems that the rooted hexagon is "distinguished", and the sequence is shifted by 1. - Petros Hadjicostas, May 26 2019

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
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; see Table 1 (p. 28).
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.
B. N. Cyvin, E. Brendsdal, J. Brunvoll, and S. J. Cyvin, A class of polygonal systems representing polycyclic conjugated hydrocarbons: Catacondensed monoheptafusenes, Monat. f. Chemie, 125 (1994), 1327-1337; see Eqs (10) and (13) on p. 1330.
S. J. Cyvin, B. N. Cyvin, J. Brunvoll, and E. Brendsdal, Enumeration and Classification of Certain Polygonal Systems Representing Polycyclic Conjugated Hydrocarbons: Annelated Catafusenes, J. Chem. Inform. Comput. Sci., 34 (1994), 1174-1180; see "Two Appendages" in Fig. 1 (p. 1176) for the unsymmetrical case (group C_s).
F. Harary and R. C. Read, The enumeration of tree-like polyhexes, Proc. Edinburgh Math. Soc. 17(2) (1970), 1-13.
Eric Weisstein's World of Mathematics, Fusene.
Wikipedia, Molecular symmetry.
Wikipedia, Point groups in three dimensions.
Wikipedia, Polyhex (mathematics).
Wikipedia, Schoenflies notation.

Cf. A002212, A045829 (auto-convolution), A002057.

a := n -> binomial(2*n+2,n+1)/(n+2) + add(binomial(2*k,k)*binomial(n-1,k-1)*(3*k-2*n-3)/(n-k+1)/(k+1),k=1..n): 0,seq(a(n),n=2..23);
# Alternative:
a := n -> (2*(n - 1)/(n + 2))*(binomial(2*n, n) / (n + 1))*hypergeom([-n-2, -n+2], [-n + 1/2], -1/4): seq(simplify(a(n)), n = 1..26); # Peter Luschny, Oct 23 2022

a[n_] = Binomial[2n+2, n+1]/(n+2) + Sum[Binomial[2k, k]*Binomial[n-1, k-1]*(3k-2n-3)/(n-k+1)/(k+1), {k, 1, n}];
a /@ Range[23] (* Jean-François Alcover, Jul 13 2011, after Maple *)
Table[SeriesCoefficient[(1/2)*(7*x^2-6*x+1+(3*x-1)*Sqrt[5*x^2-6*x+1])/x^2,{x,0,n}],{n,1,23}] (* Vaclav Kotesovec, Oct 08 2012 *)

x='x+O('x^66); concat([0],Vec((1/2)*(7*x^2-6*x+1+(3*x-1)*sqrt(5*x^2-6*x+1))/x^2)) \\ Joerg Arndt, May 04 2013

G.f.: (1/2)*(7*x^2 - 6*x + 1 + (3*x-1)*sqrt(5*x^2-6*x+1))/x^2. - Vladeta Jovovic, Jul 19 2001
a(n) = A002212(n+1) - 3*A002212(n). Convolution of A002212 without the first term with itself. - Emeric Deutsch, Jul 24 2002
a(n) = binomial(2n+2, n+1)/(n+2) + Sum_{k=1..n} binomial(2k, k)*binomial(n-1, k-1)*(3k-2n-3)/((n-k+1)*(k+1)) (n >= 2). - Emeric Deutsch, Mar 26 2004
Recurrence: (n-2)*(n+2)*a(n) = 3*(n-1)*(2*n-1)*a(n-1) - 5*(n-2)*(n-1)*a(n-2). - Vaclav Kotesovec, Oct 08 2012
a(n) ~ 5^(n+1/2)/(sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Oct 08 2012
a(n) = (2/(n+1))*Sum_{m=0..n-1} C(n+1,m)*C(2*n-2*m+2,n-m-1). - Vladimir Kruchinin Oct 18 2022
Let h(n) = hypergeom([-n-2, -n+2], [-n+1/2], -1/4) then a(n) = A002057(n-2)*h(n) = (2*(n-1)/(n+2))*CatalanNumber(n)*h(n). - Peter Luschny, Oct 23 2022

More terms from Vladeta Jovovic, Jul 19 2001

A115143 a(n) = -4binomial(2n-5, n-4)/n for n > 0 and a(0) = 1.

Original entry on oeis.org

1, -4, 2, 0, -1, -4, -14, -48, -165, -572, -2002, -7072, -25194, -90440, -326876, -1188640, -4345965, -15967980, -58929450, -218349120, -811985790, -3029594040, -11338026180, -42550029600, -160094486370, -603784920024, -2282138106804, -8643460269248, -32798844771700

Offset: 0

Wolfdieter Lang, Jan 13 2006

Previous name: Fourth convolution of A115140.
a(n+4) := - convolution ( A000108(n+1) ), n=0,1,... - Tilman Neumann, Jan 05 2009
Self-convolution of A115141. - R. J. Mathar, Sep 26 2012

Seiichi Manyama, Table of n, a(n) for n = 0..1669

Cf. A115139 - A115142, A115144 - A115149, A099376, A000108.

[1,-4,2] cat [-4*Binomial(2*n-5,n-4)/n: n in [3..30]]; // G. C. Greubel, Feb 12 2019

A115143 := n -> `if`(n=0, 1, -4*binomial(2*n-5,n-4)/n):
seq(A115143(n), n=0..28); # Peter Luschny, Feb 27 2017
A115143List := proc(m) local A, P, n; A := [1,-4,2,0]; P := [-1,0];
for n from 1 to m - 2 do P := ListTools:-PartialSums([op(P), P[-1]]);
A := [op(A), P[-1]] od; A end: A115143List(27); # Peter Luschny, Mar 26 2022

Join[{1},Table[-4*Binomial[2n-5,n-4]/n,{n,30}]] (* Harvey P. Dale, Dec 01 2017 *)
CoefficientList[Series[(1-4*x+2*x^2+(1-2*x)*Sqrt[1-4*x])/2, {x,0,30}], x] (* G. C. Greubel, Feb 12 2019 *)

my(x='x+O('x^30)); Vec((1-4*x+2*x^2 +(1-2*x)*sqrt(1-4*x))/2) \\ G. C. Greubel, Feb 12 2019

[1,-4,2] + [-4*binomial(2*n-5,n-4)/n for n in (3..30)] # G. C. Greubel, Feb 12 2019

O.g.f.: 1/c(x)^4 = P(5, x) - x*P(4, x)*c(x) with the o.g.f. c(x) := (1-sqrt(1-4*x))/(2*x) of A000108 (Catalan numbers) and the polynomials P(n, x) defined in A115139. Here P(5, x) = 1-3*x+x^2 and P(4, x) = 1-2*x.
a(n) = -C4(n-4), n>=4, with C4(n) := A002057(n) (fourth convolution of Catalan numbers). a(0)=1, a(1)=-4, a(2)=2, a(3)=0. [1, -4, 2] is row n=4 of signed A034807 (signed Lucas polynomials). See A115149 and A034807 for comments.
E.g.f.: 1 - 3*x + 1/2*x^2 - x*Q(0), where Q(k)= 1 - 2*x/(k+2 - (k+2)*(2*k+1)/(2*k+1 - (k+2)/Q(k+1))); (continued fraction). - Sergei N. Gladkovskii, Apr 28 2013
D-finite with recurrence n*(n-4)*a(n) -2*(2*n-5)*(n-3)*a(n-1)=0. - R. J. Mathar, Sep 15 2024

Simpler name from Peter Luschny, Feb 27 2017

A006659 Number of closed meander systems of order n+1 with n components.

Original entry on oeis.org

2, 12, 56, 240, 990, 4004, 16016, 63648, 251940, 994840, 3922512, 15452320, 60843510, 239519700, 942871200, 3711935040, 14615744220, 57562286760, 226760523600, 893550621600, 3522078700140, 13887053160552

Offset: 1

D. Ivanov, S. K. Lando and A. K. Zvonkin (zvonkin(AT)labri.u-bordeaux.fr)

a(n) is the total number of long interior inclines in all Dyck (n+2)-paths. An incline is a maximal subpath of like steps (all Us or all Ds); interior means it does not start or end the path; long means of length >= 2. Example: for n=1, the 5 Dyck 3-paths are shown with long interior inclines in uppercase: uuuddd, uududd, udUUdd, ududud, uuDDud and so a(1)=2. - David Callan, Jul 03 2006

a(n) is the number of corners in all parallelogram polyominoes of semiperimeter n+3. - Emeric Deutsch, Oct 09 2008

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

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000
M. Delest, J. P. Dubernard and I. Dutour, Parallelogram polyominoes and corners, J. Symbolic Computation, 20(1995),503-515.
M. P. Delest, D. Gouyou-Beauchamps and B. Vauquelin, Enumeration of parallelogram polyominoes with given bond and site parameter, Graphs and Combinatorics, 3 (1987), 325-339.
P. Di Francesco, O. Golinelli and E. Guitter, Meander, folding and arch statistics, arXiv:hep-th/9506030, 1995.
S. K. Lando and A. K. Zvonkin, Plane and projective meanders, Séries Formelles et Combinatoire Algébrique. Laboratoire Bordelais de Recherche Informatique, Université Bordeaux I, 1991, pp. 287-303. (Annotated scanned copy)
S. K. Lando and A. K. Zvonkin, Plane and projective meanders, Theoretical Computer Science Vol. 117, pp. 227-241, 1993.
Simon Plouffe, Approximations of generating functions and a few conjectures, Master's Thesis UQAM 1992, arXiv:0911.4975 [math.NT], 2009.

Cf. A002057, A001622.
Equals 2*A002694(n+1).
Cf. A005315, A005316.
A diagonal of triangle A008828.

a006659 n = 2 * a007318' (2 * n + 2) (n - 1)
-- Reinhard Zumkeller, Jun 18 2012

seq(2*binomial(2*n+2,n-1),n=1..22); # Emeric Deutsch, Oct 09 2008

f[x_] := 32/((1 + Sqrt[1 - 4x])^4*Sqrt[1 - 4x]); CoefficientList[ Series[ f[x], {x, 0, 21}], x] (* Jean-François Alcover, Dec 07 2011 *)
CoefficientList[Series[4*Exp[2x](BesselI[1,2*x]+ x(x-1)(BesselI[0,2x]+BesselI[1,2x]))/x^2,{x,0,22}],x]Table[n!,{n,0,22}] (* Stefano Spezia, May 10 2022 *)

a(n)=2*binomial(2*n+2,n-1) \\ Charles R Greathouse IV, Dec 07 2011

x='x+O('x^100); Vec(32/(sqrt(1-4*x)*(1+sqrt(1-4*x))^4)) \\ Altug Alkan, Oct 14 2015

G.f.: 32/(sqrt(1-4x)*(1+sqrt(1-4x))^4).
a(n) = (n+1) * A002057(n). - Ralf Stephan, Aug 31 2003
a(n) = 2*binomial(2n+2, n-1). - Emeric Deutsch, Oct 09 2008
a(n) = {(-56 - 30*n - 4*n^2)*a(n+1) + (8*n+12+n^2)*a(n+2), a(0)=2, a(1)=12}. - Simon Plouffe (master's thesis, 1992)
a(n) ~ 2^(2*n+3)/sqrt(n*Pi). - Charles R Greathouse IV, Dec 07 2011
E.g.f.: 4*exp(2*x)*(I_1(2*x) + x*(x - 1)*(I_0(2*x) + I_1(2*x)))/x^2, where I_n(x) is the modified Bessel function of the first kind. - Stefano Spezia, May 09 2022
From Amiram Eldar, May 15 2022: (Start)
Sum_{n>=1} 1/a(n) = 23/12 - 13*Pi/(18*sqrt(3)).
Sum_{n>=1} (-1)^(n+1)/a(n) = 53*log(phi)/(5*sqrt(5)) - 37/20, where phi is the golden ratio (A001622). (End)

A099376 An inverse Chebyshev transform of x^3.

Original entry on oeis.org

0, 1, 4, 14, 48, 165, 572, 2002, 7072, 25194, 90440, 326876, 1188640, 4345965, 15967980, 58929450, 218349120, 811985790, 3029594040, 11338026180, 42550029600, 160094486370, 603784920024, 2282138106804, 8643460269248

Offset: 0

Paul Barry, Oct 13 2004

The sequence is 0,0,0,1,0,4,0,14,0,...with zeros restored. Second binomial transform of (-1)^n*A003518(n). Second binomial transform of expansion of x^3*c(-x)^8, where c(x) is g.f. of A000108. The g.f. is transformed to x^3 under the Chebyshev transformation A(x) -> (1/(1+x^2))*A(x/(1+x^2)). For a sequence b(n), this corresponds to taking Sum_{k=0..floor(n/2)} C(n-k,k) * (-1)^k * b(n-2k), or Sum_{k=0..n} C((n+k)/2,k) * b(k) * (-1)^((n-k)/2) * (1+(-1)^(n-k))/2.

Let X_n be the set of all noncrossing set partitions of an n-element set which either do not contain {n-1,n} as a block, or which do not contain the block {n} whenever 1 and n-1 are in the same block. For n>0, (-1)^n*a(n) gives the value of the Möbius function of X_{n+2} ordered by dual refinement between the discrete and the full partition. For example, X_3 is a chain consisting of 3 elements and its Möbius function between least and greatest element therefore takes the value a(1)=0. - Henri Mühle, Jan 10 2017

G. C. Greubel, Table of n, a(n) for n = 0..1000
H. Mühle, Two Posets of Noncrossing Partitions Coming From Undesired Parking Spaces, arXiv:1701.02109 [math.CO], 2017.

Cf. A000108, A000245, A003518, A071718.

Partial sums of A026016.

[Catalan(n+2) -2*Catalan(n+1): n in [0..30]]; // G. C. Greubel, May 05 2021

Table[CatalanNumber[n+2] -2CatalanNumber[n+1], {n, 0, 30}] (* or *)
Table[4 Binomial[2#+3, #]/(#+4) &[n-1], {n, 0, 30}] (* Michael De Vlieger, Jan 10 2017, latter after Harvey P. Dale at A002057 *)

{a(n)= if(n<1, 0, n++; 2* binomial(2*n, n-2)/n)} /* Michael Somos, Apr 11 2007 */

[catalan_number(n+2) -2*catalan_number(n+1) for n in (0..30)] # G. C. Greubel, May 05 2021

G.f.: (1-2*x)^4*( sqrt((1+2*x)/(1-2*x)) - 1)^8/(256*x^5).
a(n) = Sum_{k=0..n} (k+1)*C(n, (n-k)/2)*(-1)^k*( C(3, k) -3*C(2, k) +3*C(1, k) -C(0, k) )*(1+(-1)^(n-k))/(n+k+2).
a(n) = A002057(n-1). - Michael Somos, Jul 31 2005
Given an ellipse with eccentricity e and major and minor axis a and b respectively, then ((a-b)/ (a+b))^2 = 1*(e/2)^4 +4*(e/2)^6 +14*(e/2)^8 +48*(e/2)^10 + ... - Michael Somos, Apr 11 2007
E.g.f.: exp(2x)*(Bessel_I(1,2x) - Bessel_I(3,2x)). - Paul Barry, Jun 04 2007
D-finite with recurrence (n+3)*(n-1)*a(n) -2*n*(2*n+1)*a(n-1)=0. - R. J. Mathar, Sep 26 2012
a(n) = A000108(n+2) - 2*A000108(n+1) for n>0. - Henri Mühle, Jan 10 2017, corrected Sep 25 2021
G.f.: ( (1-2*x)*c(x) - (1-x) )/x^2, where c(x) is the gf of A000108. - G. C. Greubel, May 05 2021
From Peter Bala, Aug 30 2023: (Start)
a(n) = 2*n/((n+2)*(n+3)) * binomial(2*n+2, n+1).
a(n) = 2*Sum_{k = 0..n-1} 1/(n+1)*binomial(n+1, k)*binomial(n+1, k+2). (End)

A102071 Pairwise sums of general ballot numbers (A002026).

Original entry on oeis.org

1, 3, 7, 17, 42, 106, 272, 708, 1865, 4963, 13323, 36037, 98123, 268737, 739833, 2046207, 5682915, 15842505, 44315637, 124348275, 349911204, 987212856, 2791964574, 7913642086, 22477090679, 63964370301, 182353459733, 520735012027, 1489362193002, 4266018891562, 12236183875496, 35142703099692, 101055137177563

Offset: 1

Ralf Stephan, Dec 30 2004

G. C. Greubel, Table of n, a(n) for n = 1..1000
Gennady Eremin, Naturalized bracket row and Motzkin triangle, arXiv:2004.09866 [math.CO], 2020. See Table 2.

First differences of A005554. Partial sums of A026269. 3rd column of A348840.

Cf. A000108, A001006, A002057.

CoefficientList[Series[(4x(1+x))/(1-x+Sqrt[1-2x-3x^2])^2,{x,0,40}],x] (* Harvey P. Dale, Feb 26 2013 *)

a(n):=1/n*sum((binomial(j,n-1-j)+4*binomial(j,n-2-j)+3*binomial(j,n-3-j))*binomial(n,j),j,0,n); /* Vladimir Kruchinin, Mar 08 2016 */

z='z+O('z^66); Vec(4*z*(1+z)/(1-z+sqrt(1-2*z-3*z^2))^2) \\ Joerg Arndt, Mar 08 2016

G.f.: (4*x*(1+x))/(1-x+sqrt(1-2*x-3*x^2))^2.
a(n) = (1/n) * Sum_{j=0..n} ((binomial(j,n-1-j)+4*binomial(j,n-2-j) + 3*binomial(j,n-3-j))*binomial(n,j)). - Vladimir Kruchinin, Mar 08 2016
a(n) ~ 4*3^(n+1/2)/(sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Mar 08 2016
a(n) = A001006(n+1) - A001006(n-1). - Gennady Eremin, Sep 23 2021
D-finite with recurrence (n+3)*a(n) + (-3*n-5)*a(n-1) + (-n+3)*a(n-2) + 3*(n-3)*a(n-3) = 0. - R. J. Mathar, Nov 01 2021
From Peter Bala, Feb 02 2024: (Start)
a(n) = Sum_{k = 0..n} (-1)^(n-k)*binomial(n,k)*A002057(k).
G.f.: x/(1 + x)*c(x/(1 + x))^4, where c(x) = (1 - sqrt(1 - 4*x))/(2*x) is the g.f. of the Catalan numbers A000108. (End)

A003446 Number of nonequivalent dissections of a polygon into n triangles by nonintersecting diagonals rooted at a cell up to rotation and reflection.

Original entry on oeis.org

0, 1, 1, 2, 6, 16, 52, 170, 579, 1996, 7021, 24892, 89214, 321994, 1170282, 4277352, 15715249, 57999700, 214939846, 799478680, 2983699498, 11169391168, 41929537871, 157807451672, 595340479694, 2250901216266, 8527700012092, 32369067177176

Offset: 0

N. J. A. Sloane

Original name: Triangulated (n+2)-gons rooted at one of the triangles.

Also, the total number of atom-rooted polyenoids. - Sean A. Irvine, Oct 05 2015

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..200
S. J. Cyvin, J. Brunvoll, E. Brendsdal, B. N. Cyvin and E. K. Lloyd, Enumeration of polyene hydrocarbons: a complete mathematical solution, J. Chem. Inf. Comput. Sci., 35 (1995) 743-751.
S. J. Cyvin, J. Brunvoll, E. Brendsdal, B. N. Cyvin and E. K. Lloyd, Enumeration of polyene hydrocarbons: a complete mathematical solution, J. Chem. Inf. Comput. Sci., 35 (1995) 743-751. [Annotated scanned copy]
F. Harary, E. M. Palmer, R. C. Read, On the cell-growth problem for arbitrary polygons, computer printout, circa 1974
F. Harary, E. M. Palmer and R. C. Read, On the cell-growth problem for arbitrary polygons, Discr. Math. 11 (1975), 371-389.
P. Lisonek, Closed forms for the number of polygon dissections, Journal of Symbolic Computation 20 (1995), 595-601.
Ronald C. Read, On general dissections of a polygon, Aequat. math. 18 (1978) 370-388, Table 1.
P. K. Stockmeyer, The charm bracelet problem and its applications, pp. 339-349 of Graphs and Combinatorics (Washington, Jun 1973), Ed. by R. A. Bari and F. Harary. Lect. Notes Math., Vol. 406. Springer-Verlag, 1974.
P. J. Stockmeyer, The charm bracelet problem and its applications, pp. 339-349 of Graphs and Combinatorics (Washington, Jun 1973), Ed. by R. A. Bari and F. Harary. Lect. Notes Math., Vol. 406. Springer-Verlag, 1974. [Scanned annotated and corrected copy]

Column k=3 of A295259.

c[x_] = (1 - Sqrt[1 - 4*x])/(2*x); d[x_] = 1 + x*c[x^2]; f[x_] = (x/6)*(c[x]^3 + 2*c[x^3] + 3*d[x]*c[x^2]); CoefficientList[ Series[ f[x], {x, 0, 27}], x] (* Jean-François Alcover, Sep 30 2011, after g.f. *)

Let c(x) = (1-sqrt(1-4*x))/(2*x) = g.f. for Catalan numbers (A000108), let d(x) = 1+x*c(x^2). Then g.f. is (x/6)*(c^3+2*subs(x=x^3, c)+3*d*subs(x=x^2, c)).
Recurrence: n*(n+1)*(n+2)*(12*n^10 - 396*n^9 + 5713*n^8 - 47417*n^7 + 250708*n^6 - 883176*n^5 + 2104831*n^4 - 3368071*n^3 + 3489712*n^2 - 2133004*n + 587808)*a(n) = 2*(n-1)*n*(n+1)*(24*n^10 - 756*n^9 + 10262*n^8 - 78647*n^7 + 374743*n^6 - 1154043*n^5 + 2323495*n^4 - 3057578*n^3 + 2632172*n^2 - 1456776*n + 412560)*a(n-1) + 4*(n-1)*n*(12*n^11 - 384*n^10 + 5377*n^9 - 43234*n^8 + 219811*n^7 - 731024*n^6 + 1576767*n^5 - 2055172*n^4 + 1195025*n^3 + 527398*n^2 - 1223056*n + 534240)*a(n-2) - 2*(72*n^13 - 2484*n^12 + 37950*n^11 - 339019*n^10 + 1971954*n^9 - 7887993*n^8 + 22425262*n^7 - 46437513*n^6 + 71577166*n^5 - 83189763*n^4 + 71509420*n^3 - 41716412*n^2 + 13543200*n - 1451520)*a(n-3) - 4*(n-1)*n*(2*n - 7)*(24*n^10 - 756*n^9 + 10262*n^8 - 78647*n^7 + 374743*n^6 - 1154043*n^5 + 2323495*n^4 - 3057578*n^3 + 2632172*n^2 - 1456776*n + 412560)*a(n-4) - 8*(n-1)*(2*n - 9)*(12*n^11 - 384*n^10 + 5377*n^9 - 43234*n^8 + 219811*n^7 - 731024*n^6 + 1576767*n^5 - 2055172*n^4 + 1195025*n^3 + 527398*n^2 - 1223056*n + 534240)*a(n-5) + 16*(n-6)*(2*n - 11)*(2*n - 9)*(12*n^10 - 276*n^9 + 2689*n^8 - 14529*n^7 + 48009*n^6 - 101629*n^5 + 142510*n^4 - 137838*n^3 + 93836*n^2 - 39760*n + 6720)*a(n-6). - Vaclav Kotesovec, Aug 13 2013
a(n) ~ 4^n/(2*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Aug 13 2013

Name edited by Andrew Howroyd, Nov 20 2017

A090749 a(n) = 12 * C(2n+1,n-5) / (n+7).

Original entry on oeis.org

1, 12, 90, 544, 2907, 14364, 67298, 303600, 1332045, 5722860, 24192090, 100975680, 417225900, 1709984304, 6962078952, 28192122176, 113649492522, 456442180920, 1827459250276, 7297426411968, 29075683360185, 115631433392020, 459124809056550, 1820529677650320, 7210477496434485

Offset: 5

Philippe Deléham, Feb 15 2004

Also a diagonal of A059365 and A009766. See also A000108, A002057, A003517, A003518, A003519.

Number of standard tableaux of shape (n+6,n-5). - Emeric Deutsch, May 30 2004

G. C. Greubel, Table of n, a(n) for n = 5..1000
Daniel Birmajer, Juan B. Gil, and Michael D. Weiner, Bounce statistics for rational lattice paths, arXiv:1707.09918 [math.CO], 2017, p. 9.
Richard K. Guy, Letter to N. J. A. Sloane, May 1990.
Richard K. Guy, Catwalks, sandsteps and Pascal pyramids, J. Integer Sequences, Vol. 3 (2000), Article 00.1.6.
V. E. Hoggatt, Jr., 7-page typed letter to N. J. A. Sloane with suggestions for new sequences, circa 1977.

Cf. A001622, A009766, A039598, A059365, A214292.

Cf. A000108, A002057, A003517, A003518, A003519.

Table[12*Binomial[2*n + 1, n - 5]/(n + 7), {n,5,50}] (* G. C. Greubel, Feb 07 2017 *)

for(n=5,50, print1(12*binomial(2*n+1,n-5)/(n+7), ", ")) \\ G. C. Greubel, Feb 07 2017

a(n) = A039598(n, 5) = A033184(n+7, 12).
G.f.: x^5*C(x)^12 with C(x) g.f. of A000108(Catalan).
Let A be the Toeplitz matrix of order n defined by: A[i,i-1]=-1, A[i,j]=Catalan(j-i), (i<=j), and A[i,j]=0, otherwise. Then, for n>=11, a(n-6)=(-1)^(n-11)*coeff(charpoly(A,x),x^11). - Milan Janjic, Jul 08 2010
a(n) = A214292(2*n,n-6) for n > 5. - Reinhard Zumkeller, Jul 12 2012
From Karol A. Penson, Nov 21 2016: (Start)
O.g.f.: z^5 * 4^6/(1+sqrt(1-4*z))^12.
Recurrence: (-4*(n-5)^2-58*n+80)*a(n+1)-(-n^2-6*n+27)*a(n+2)=0, a(0),a(1),a(2),a(3),a(4)=0,a(5)=1,a(6)=12, n>=5.
Asymptotics: (-903+24*n)*4^n*sqrt(1/n)/(sqrt(Pi)*n^2).
Integral representation as n-th moment of a signed function W(x) on x=(0,4),in Maple notation: a(n+5)=int(x^n*W(x),x=0..4),n=0,1,..., where W(x)=(256/231)*sqrt(4-x)*JacobiP(5, 1/2, 1/2, (1/2)*x-1)*x^(11/2)/Pi and JacobiP are Jacobi polynomials. Note that W(0)=W(4)=0. (End).
From Ilya Gutkovskiy, Nov 21 2016: (Start)
E.g.f.: 6*exp(2*x)*BesselI(6,2*x)/x.
a(n) ~ 3*2^(2*n+3)/(sqrt(Pi)*n^(3/2)). (End)
From Amiram Eldar, Sep 26 2022: (Start)
Sum_{n>=5} 1/a(n) = 88699/15120 - 71*Pi/(27*sqrt(3)).
Sum_{n>=5} (-1)^(n+1)/a(n) = 102638*log(phi)/(75*sqrt(5)) - 22194839/75600, where phi is the golden ratio (A001622). (End)

Missing term 113649492522 inserted by Ilya Gutkovskiy, Dec 07 2016

A119245 Triangle, read by rows, defined by: T(n,k) = (4k+1)binomial(2n+1, n-2k)/(2n+1) for n >= 2k >= 0.

Original entry on oeis.org

1, 1, 2, 1, 5, 5, 14, 20, 1, 42, 75, 9, 132, 275, 54, 1, 429, 1001, 273, 13, 1430, 3640, 1260, 104, 1, 4862, 13260, 5508, 663, 17, 16796, 48450, 23256, 3705, 170, 1, 58786, 177650, 95931, 19019, 1309, 21, 208012, 653752, 389367, 92092, 8602, 252, 1

Offset: 0

Paul D. Hanna, May 10 2006

Closely related to triangle A118919.
Row n contains 1+floor(n/2) terms.
From Peter Bala, Mar 20 2009: (Start)
Combinatorial interpretations of T(n,k):
1) The number of standard tableaux of shape (n-2*k,n+2*k).
2) The entries in column k are (with an offset of 2*k) the number of n-th generation vertices in the tree of sequences with unit increase labeled by 4*k. See [Sunik, Theorem 4]. (End)

			Triangle begins:
     1;
     1;
     2,     1;
     5,     5;
    14,    20,    1;
    42,    75,    9;
   132,   275,   54,   1;
   429,  1001,  273,  13;
  1430,  3640, 1260, 104,  1;
  4862, 13260, 5508, 663, 17; ...

Zoran Sunic, Self describing sequences and the Catalan family tree, Elect. J. Combin., 10 (No. 1, 2003). - _Peter Bala_, Mar 20 2009

Cf. A119244 (eigenvector), A088218, A000108, A000344, A001392; A118919 (variant), A158483; A002057, A002894.

f1 = (1-Sqrt[1-4*x])/(2*x);
DeleteCases[CoefficientList[Normal@Series[f1/(1 - x^2*y*f1^4),{x,0,10},{y,0,5}],{x,y}],0,Infinity]//TableForm  (* Bradley Klee, Feb 26 2018 *)
Table[(1+4*k)/(n+1+2*k)*Binomial[2*n,n+2*k],{n,0,10},{k,0,Floor[n/2]}]//TableForm (* Bradley Klee, Feb 26 2018 *)

T(n,k)=(4*k+1)*binomial(2*n+1,n-2*k)/(2*n+1)

G.f.: A(x,y) = f/(1-x^2*y*f^4), where f=(1-sqrt(1-4*x))/(2*x) is the Catalan g.f. (A000108).
Row sums equal A088218(n) = C(2*n-1,n).
T(n,0) = A000108(n) (the Catalan numbers).
T(n,1) = A000344(n).
T(n,2) = A001392(n).
Sum_{k=0..floor(n/2)} k*T(n,k) = A000346(n-2).
Eigenvector is defined by: A119244(n) = Sum_{k=0..[n\2]} T(n,k)*A119244(k).
...
T(n,k) = (4*k+1)/(n+2*k+1)*binomial(2*n,n+2*k). Compare with A158483. - Peter Bala, Mar 20 2009
T(n,k) = A039599(n, 2*k). - Johannes W. Meijer, Sep 04 2013
A002894(n) = Sum_{k=0..floor(n/2)} (binomial(2k,k)^2)*(4^(n-2*k))*T(n,k). - Bradley Klee, Feb 26 2018

A120989 Level of the first leaf (in preorder traversal) of a binary tree, summed over all binary trees with n edges. A binary tree is a rooted tree in which each vertex has at most two children and each child of a vertex is designated as its left or right child.

Original entry on oeis.org

2, 9, 34, 123, 440, 1573, 5642, 20332, 73644, 268090, 980628, 3603065, 13293540, 49234605, 182991450, 682341000, 2551955340, 9570762990, 35985909180, 135628219350, 512302356384, 1939078493154, 7353556121924, 27936898370248, 106313496846200

Offset: 1

Emeric Deutsch, Jul 30 2006

a(n) is the number of lattice paths from (0,0) to (n+2,n+2) using E(1,0) and N(0,1) as steps that have exactly two E steps below subdiagonal y = x-1. - Ran Pan, Feb 01 2016

a(n) is the number of permutations pi of [n+3] such that s(pi)=p456...(n+3), where s is West's stack-sorting map and p=132. The same statement is true if p=231 or p=312. - Colin Defant, Jan 14 2019

			a(1)=2 because for each of the trees / and \ the level of the first leaf is 1.

Ran Pan and Jeffrey B. Remmel, Paired patterns in lattice paths, arXiv:1601.07988 [math.CO], 2016-2017.

Cf. A120988, A002057.

[2*n*(7*n+13)*Binomial(2*n+1,n)/((n+2)*(n+3)*(n+4)): n in [1..30]]; // Vincenzo Librandi, Feb 01 2016

a:=n->2*n*(7*n+13)*binomial(2*n+1,n)/(n+2)/(n+3)/(n+4): seq(a(n),n=1..27);

Table[2 n (7 n + 13) Binomial[2 n + 1, n] / ((n + 2) (n + 3) (n + 4)), {n, 30}] (* Vincenzo Librandi, Feb 01 2016 *)

a(n)=2*n*(7*n+13)*binomial(2*n+1,n)/prod(i=2,4,n+i) \\ Charles R Greathouse IV, Feb 01 2016

a(n) = Sum_{k=1..n} k*A120988(n,k).
a(n) = 2*n*(7n+13)*binomial(2n+1,n)/((n+2)(n+3)(n+4)).
G.f.: z*(1+C)*C^4, where C = (1-sqrt(1-4*z))/(2z) is the Catalan function.
G.f.: 2*(1+2*z-sqrt(1-4*z))/(1-2*z+sqrt(1-4*z))^2.
D-finite with recurrence -(n-1)*(7*n+6)*(n+4)*a(n) +2*n*(7*n+13)*(2*n+1)*a(n-1)=0. - R. J. Mathar, Aug 22 2016
a(n) ~ c*4^n*n^(-3/2), with c = 28/sqrt(Pi). - Stefano Spezia, Oct 19 2023

A262543 Number of rooted asymmetrical polyenoids of type U_n* having n edges.

Original entry on oeis.org

0, 2, 4, 10, 28, 84, 264, 858, 2860, 9724, 33592, 117572, 416024, 1485800, 5348880, 19389690, 70715340, 259289580, 955277400, 3534526380, 13128240840, 48932534040, 182965127280, 686119227300, 2579808294648, 9723892802904, 36734706144304, 139067101832008, 527495903500720

Offset: 1

N. J. A. Sloane, Oct 06 2015

Essentially the same as A068875 and A002420. See those entries for much more information.

			G.f. = 2*x^2 + 4*x^3 + 10*x^4 + 28*x^5 + 84*x^6 + 264*x^7 + 858*x^8 + ...

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
S. J. Cyvin, J. Brunvoll, E. Brendsdal, B. N. Cyvin and E. K. Lloyd, Enumeration of polyene hydrocarbons: a complete mathematical solution, J. Chem. Inf. Comput. Sci., 35 (1995) 743-751.
S. J. Cyvin, J. Brunvoll, E. Brendsdal, B. N. Cyvin and E. K. Lloyd, Enumeration of polyene hydrocarbons: a complete mathematical solution, J. Chem. Inf. Comput. Sci., 35 (1995) 743-751. [Annotated scanned copy]

Cf. A068875, A002420.

[0] cat [2*Catalan(n-1): n in [2..40]]; // Vincenzo Librandi, Feb 28 2017

A262543List := proc(m) local A, P, n; A := [0, 2]; P := [2];
for n from 1 to m - 2 do P := ListTools:-PartialSums([op(P), P[-1]]);
A := [op(A), P[-1]] od; A end: A262543List(29); # Peter Luschny, Mar 24 2022

Table[If[n==1,0,2 CatalanNumber[n-1]], {n,1,26}] (* Peter Luschny, Feb 27 2017 *)
Join[{0}, Rest[CoefficientList[Series[2 (1 - Sqrt[1 - 4 x])/(2 x), {x, 0, 30}], x]]] (* Vincenzo Librandi, Feb 28 2017 *)

a(n)=if(n>1, 2*binomial(2*n-2,n-1)/n, 0) \\ Charles R Greathouse IV, Feb 28 2017

From Gennady Eremin, Feb 22 2021: (Start)
G.f.: 1 - 2*x - sqrt(1 - 4*x).
G.f.: x*(A(x) - 1), where A(x) is the g.f. of A068875. (End)

cp's OEIS Frontend

A045445 Number of nonisomorphic systems of catafusenes for the unsymmetrical schemes (group C_s) with two appendages (see references for precise definition).

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A115143 a(n) = -4binomial(2n-5, n-4)/n for n > 0 and a(0) = 1.

A119245 Triangle, read by rows, defined by: T(n,k) = (4k+1)binomial(2n+1, n-2k)/(2n+1) for n >= 2k >= 0.