%I A090344
%S A090344 1,1,2,3,6,11,23,47,102,221,493,1105,2516,5763,13328,30995,72556,170655,
%T A090344 403351,957135,2279948,5449013,13063596,31406517,75701508,182902337,
%U A090344 442885683,1074604289,2612341856,6361782007,15518343597,37912613631
%N A090344 Number of Motzkin paths of length n with no level steps at odd level.
%C A090344 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
%C A090344 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
%F A090344 G.f.=[1-z-sqrt(1-2z-3z^2+4z^3)]/[2z^2(1-z)].
%F A090344 (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
%F A090344 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
%F A090344 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
%F A090344 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]
%e A090344 a(3)=3 because we have HHH, HUD and UDH, where U=(1,1), D=(1,-1) and 
               H=(1,0).
%p A090344 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);
%Y A090344 Cf. A001006, A098474, A124497, A124344.
%Y A090344 Sequence in context: A036592 A001190 A036656 this_sequence A130131 A123465 
               A000055
%Y A090344 Adjacent sequences: A090341 A090342 A090343 this_sequence A090345 A090346 
               A090347
%K A090344 nonn
%O A090344 0,3
%A A090344 Emeric Deutsch (deutsch(AT)duke.poly.edu), Jan 28 2004