%I A117641
%S A117641 1,0,1,3,11,42,167,684,2867,12240,53043,232731,1031829,4615542,20805081,
%T A117641 94410363,430945739,1977366192,9115261211,42195093993,196060049129,
%U A117641 914110333422,4275222950221,20051858039718,94294269673861
%N A117641 Number of 3-Motzkin paths with no level steps at height 0.
%C A117641 Hankel transform of this sequence forms A000012 = [1,1,1,1,1,...] . - 
               Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Oct 24 2007
%F A117641 Generating function = (1+3z-sqrt(1-6z+5z^2))/(2z^2+6z).
%F A117641 G.f. as continued fraction is 1/(1-0*x-x^2/(1-3*x-x^2/(1-3*x-x^2/(1-3*x-x^2/
               (..... [From Paul Barry (pbarry(AT)wit.ie), Dec 02 2008]
%F A117641 a(n)= A126970(n,0). [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), 
               Nov 24 2009]
%F A117641 a(n)= Sum_{k, 0<=k<=n} A091965(n,k)*(-3)^k. [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), 
               Nov 28 2009]
%e A117641 The a(4) = 11 paths are UUDD, UDUD and 9 of the form UXYD where each 
               of X and Y are level steps in any of three colors.
%t A117641 CoefficientList[ Series[(1 + 3x - Sqrt[1 - 6x + 5x^2])/(2x^2 + 6x), {x, 
               0, 25}], x] (* Robert G. Wilson v *)
%Y A117641 Cf. A000957, A001006, A002212, A005043, A097331, A000108.
%Y A117641 Sequence in context: A122368 A032443 A143464 this_sequence A084782 A149068 
               A151088
%Y A117641 Adjacent sequences: A117638 A117639 A117640 this_sequence A117642 A117643 
               A117644
%K A117641 easy,nonn,new
%O A117641 0,4
%A A117641 Louis Shapiro (lshapiro(AT)howard.edu), Apr 10 2006
%E A117641 More terms from Robert G. Wilson v (rgwv(at)rgwv.com), Apr 12 2006