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

1,3

Comments

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

Links

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.

Formula

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

Maple

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

Mathematica

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 *)

Prog

(PARI) 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

Crossrefs

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

Sequence in context: A026873 A081179 A026866 * A026884 A289801 A359920

Adjacent sequences: A045442 A045443 A045444 * A045446 A045447 A045448

Keyword

nonn,easy

Author

N. J. A. Sloane

Extensions

More terms from Vladeta Jovovic, Jul 19 2001

Status

approved