0,3

a(n) = number of Motzkin paths of length n that avoid UF. Example: a(3) counts FFF, UDF, FUD but not UFD. - David Callan (callan(AT)stat.wisc.edu), Jul 15 2004

Also, number of 1-2 trees with n edges and with thinning limbs. A 1-2 tree is an ordered tree with vertices of outdegree at most 2. A rooted tree with thinning limbs is such that if a node has k children, all its children have at most k children. - Emeric Deutsch and Louis Shapiro (deutsch(AT)duke.poly.edu, lshapiro(AT)Howard.edu), Nov 04 2006

G.f.=[1-z-sqrt(1-2z-3z^2+4z^3)]/[2z^2(1-z)].

(n+2)*a(n)-(2*n+2)*a(n-1)-(3*n-4)*a(n-2)+(4*n-6)*a(n-3) = 0. - Vladeta Jovovic (vladeta(AT)eunet.rs), Sep 11 2004

a(n)=sum{k=0..floor(n/2), binomial(n-k, k)binomial(2k, k)/(k+1)}. - Paul Barry (pbarry(AT)wit.ie), Nov 13 2004

a(n) = 1 + sum_k a(k-1)a(n-k-1), starting from a(n)=0 for n negative. - Henry Bottomley (se16(AT)btinternet.com), Feb 22 2005

G.f.: 1/(1-x-x^2/(1-x^2/(1-x-x^2/(1-x^2/(1-x-x^2/(1-x^2/(1-... (continued fraction). [From Paul Barry (pbarry(AT)wit.ie), Apr 08 2009]

a(3)=3 because we have HHH, HUD and UDH, where U=(1,1), D=(1,-1) and H=(1,0).

C:=x->(1-sqrt(1-4*x))/2/x: G:=C(z^2/(1-z))/(1-z): Gser:=series(G, z=0, 40): seq(coeff(Gser, z, n), n=0..36);

Cf. A001006, A098474, A124497, A124344.

Sequence in context: A036592 A001190 A036656 this_sequence A130131 A123465 A000055

Adjacent sequences: A090341 A090342 A090343 this_sequence A090345 A090346 A090347

nonn

Emeric Deutsch (deutsch(AT)duke.poly.edu), Jan 28 2004