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A045445
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Catafusenes (see references for precise definition).
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3
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0, 1, 6, 29, 132, 590, 2628, 11732, 52608, 237129, 1074510, 4893801, 22395420, 102943815, 475139070, 2201301575, 10234016880, 47731093715, 223273611810, 1047265325255, 4924606035900, 23211459517120, 109642275853176
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,3
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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
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REFERENCES
| B. N. Cyvin et al., A class of polygonal systems representing polycyclic conjugated hydrocarbons ..., Monat. f. Chemie, 125 (1994), 1327-1337 (see U(x)).
S. J. Cyvin et al., Enumeration and classification of certain polygonal systems... : annelated catafusenes, J. Chem. Inform. Comput. Sci., 34 (1994), 1174-1180.
F. Harary and R. C. Read, The enumeration of tree-like polyhexes, Proc. Edinb. Math. Soc. (2) 17 (1970), 1-13.
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FORMULA
| G.f.: (1/2)*(7*x^2-6*x+1+(3*x-1)*sqrt(5*x^2-6*x+1))/x^2.
A045445(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(binomial(2k, k)*binomial(n-1, k-1)*(3k-2n-3)/[(n-k+1)(k+1)], k=1..n) (n>=2). - Emeric Deutsch, Mar 26 2004
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MAPLE
| a := n->binomial(2*n+2, n+1)/(n+2)+sum(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);
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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] (* From Jean-François Alcover, Jul 13 2011, after Maple *)
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CROSSREFS
| Cf. A002212.
Sequence in context: A026873 A081179 A026866 * A026884 A110311 A030221
Adjacent sequences: A045442 A045443 A045444 * A045446 A045447 A045448
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KEYWORD
| nonn
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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EXTENSIONS
| G.f. and more terms from Vladeta Jovovic (vladeta(AT)eunet.rs), Jul 19 2001
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