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

%I A045445 #68 Feb 16 2025 08:32:38
%S A045445 0,1,6,29,132,590,2628,11732,52608,237129,1074510,4893801,22395420,
%T A045445 102943815,475139070,2201301575,10234016880,47731093715,223273611810,
%U A045445 1047265325255,4924606035900,23211459517120,109642275853176,518959629394294,2460993383491632,11691102386417575
%N A045445 Number of nonisomorphic systems of catafusenes for the unsymmetrical schemes (group C_s) with two appendages (see references for precise definition).
%C A045445 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
%C A045445 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
%H A045445 Vincenzo Librandi, <a href="/A045445/b045445.txt">Table of n, a(n) for n = 1..1000</a>
%H A045445 S. J. Cyvin, J. Brunvoll, and B. N. Cyvin, <a href="https://doi.org/10.1007/BF01172927">Harary-Read numbers for catafusenes: Complete classification according to symmetry</a>, Journal of Mathematical Chemistry 9(1) (1992), 19-31; see Table 1 (p. 28).
%H A045445 S. J. Cyvin and J. Brunvoll, <a href="https://doi.org/10.1007/BF01172928">Generating functions for the Harary-Read numbers classified according to symmetry</a>, Journal of Mathematical Chemistry 9(1) (1992), 33-38.
%H A045445 B. N. Cyvin, E. Brendsdal, J. Brunvoll, and S. J. Cyvin, <a href="http://dx.doi.org/10.1007/BF00811082">A class of polygonal systems representing polycyclic conjugated hydrocarbons: Catacondensed monoheptafusenes</a>, Monat. f. Chemie, 125 (1994), 1327-1337; see Eqs (10) and (13) on p. 1330.
%H A045445 S. J. Cyvin, B. N. Cyvin, J. Brunvoll, and E. Brendsdal, <a href="http://dx.doi.org/10.1021/ci00021a026">Enumeration and Classification of Certain Polygonal Systems Representing Polycyclic Conjugated Hydrocarbons: Annelated Catafusenes</a>, J. Chem. Inform. Comput. Sci., 34 (1994), 1174-1180; see "Two Appendages" in Fig. 1 (p. 1176) for the unsymmetrical case (group C_s).
%H A045445 F. Harary and R. C. Read, <a href="https://doi.org/10.1017/S0013091500009135">The enumeration of tree-like polyhexes</a>, Proc. Edinburgh Math. Soc. 17(2) (1970), 1-13.
%H A045445 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Fusene.html">Fusene</a>.
%H A045445 Wikipedia, <a href="https://en.wikipedia.org/wiki/Molecular_symmetry">Molecular symmetry</a>.
%H A045445 Wikipedia, <a href="https://en.wikipedia.org/wiki/Point_groups_in_three_dimensions">Point groups in three dimensions</a>.
%H A045445 Wikipedia, <a href="https://en.wikipedia.org/wiki/Polyhex_(mathematics)">Polyhex (mathematics)</a>.
%H A045445 Wikipedia, <a href="https://en.wikipedia.org/wiki/Schoenflies_notation">Schoenflies notation</a>.
%F A045445 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
%F A045445 a(n) = A002212(n+1) - 3*A002212(n). Convolution of A002212 without the first term with itself. - _Emeric Deutsch_, Jul 24 2002
%F A045445 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
%F A045445 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
%F A045445 a(n) ~ 5^(n+1/2)/(sqrt(Pi)*n^(3/2)). - _Vaclav Kotesovec_, Oct 08 2012
%F A045445 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
%F A045445 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
%p A045445 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);
%p A045445 # Alternative:
%p A045445 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
%t A045445 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}];
%t A045445 a /@ Range[23] (* _Jean-François Alcover_, Jul 13 2011, after Maple *)
%t A045445 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 *)
%o A045445 (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
%Y A045445 Cf. A002212, A045829 (auto-convolution), A002057.
%K A045445 nonn,easy
%O A045445 1,3
%A A045445 _N. J. A. Sloane_
%E A045445 More terms from _Vladeta Jovovic_, Jul 19 2001