1,3

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

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.

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 (deutsch(AT)duke.poly.edu), 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 (deutsch(AT)duke.poly.edu), Mar 26 2004

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);

Cf. A002212.

Sequence in context: A026873 A081179 A026866 this_sequence A026884 A110311 A030221

Adjacent sequences: A045442 A045443 A045444 this_sequence A045446 A045447 A045448

nonn

N. J. A. Sloane (njas(AT)research.att.com).

G.f. and more terms from Vladeta Jovovic (vladeta(AT)eunet.rs), Jul 19 2001