%I A045445
%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
%N A045445 Catafusenes (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 (deutsch(AT)duke.poly.edu), 
               Mar 26 2004
%D A045445 B. N. Cyvin et al., A class of polygonal systems representing polycyclic 
               conjugated hydrocarbons ..., Monat. f. Chemie, 125 (1994), 1327-1337 
               (see U(x)).
%D A045445 S. J. Cyvin et al., Enumeration and classification of certain polygonal 
               systems... : annelated catafusenes, J. Chem. Inform. Comput. Sci., 
               34 (1994), 1174-1180.
%D A045445 F. Harary and R. C. Read, The enumeration of tree-like polyhexes, Proc. 
               Edinb. Math. Soc. (2) 17 (1970), 1-13.
%F A045445 G.f.: (1/2)*(7*x^2-6*x+1+(3*x-1)*sqrt(5*x^2-6*x+1))/x^2.
%F A045445 A045445(n)=A002212(n+1)-3*A002212(n). Convolution of A002212 without 
               the first term with itself. - Emeric Deutsch (deutsch(AT)duke.poly.edu), 
               Jul 24 2002
%F 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) (n>=2). - Emeric Deutsch (deutsch(AT)duke.poly.edu), 
               Mar 26 2004
%p A045445 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);
%Y A045445 Cf. A002212.
%Y A045445 Sequence in context: A026873 A081179 A026866 this_sequence A026884 A110311 
               A030221
%Y A045445 Adjacent sequences: A045442 A045443 A045444 this_sequence A045446 A045447 
               A045448
%K A045445 nonn
%O A045445 1,3
%A A045445 N. J. A. Sloane (njas(AT)research.att.com).
%E A045445 G.f. and more terms from Vladeta Jovovic (vladeta(AT)eunet.rs), Jul 19 
               2001