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
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.
Programs
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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
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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 *) -
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
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
Extensions
More terms from Vladeta Jovovic, Jul 19 2001
Comments