%I A095981
%S A095981 0,0,1,2,5,11,26,61,147,357,879,2183,5471,13811,35100,89724,230562,
%T A095981 595237,1543191,4016038,10487553,27473602,72178312,190127740,502044221,
%U A095981 1328667241,3523684572,9363119781,24924679832,66461841934,177501561659
%N A095981 Number of plateau-free Motzkin paths of length n.
%C A095981 A plateau in a Motzkin path is a sequence of contiguous flatsteps that 
               is either the entire path or of length >=1 and preceded by an up 
               step and followed by a down step. a(n) = number of plateau-free Motzkin 
               paths of length n.
%F A095981 a(n) = a(n-1) + a(n-2) + 1 + a(2)(1 + a(n-4) )+a(3)(1 + a(n-5)) + ... 
               + a(n-2)(1 + a(0)) for n>=3. This recurrence counts plateau-free 
               Motzkin n-paths by location of first return to ground level. G.f.: 
               (-1 + 2*x + x^2 - x^3 + (1 - 4*x + 2*x^2 + 6*x^3 - 7*x^4 + 2*x^5 
               + x^6)^(1/2))/(2*(-1 + x)*x^2). Satisfies x^2*(1-x)*A(x)^2-(1-2*x-x^2+x^3)*A(x)+x^2=0.
%e A095981 The middle two steps of UFFD form a plateau and a(4) counts the 5 paths 
               FFUD,FUDF,UDFF,UDUD,UUDD.
%t A095981 a[0] = 0; a[1] = 0; a[2] = 1; a[n_]/;n>=3 := a[n] = a[n-1] + a[n-2] + 
               1 + Sum[(a[k])(1+a[n-2-k]), {k, 2, n-2}]; Table[a[n], {n, 0, 15}]
%Y A095981 Sequence in context: A064416 A006138 A124217 this_sequence A082397 A051286 
               A025245
%Y A095981 Adjacent sequences: A095978 A095979 A095980 this_sequence A095982 A095983 
               A095984
%K A095981 nonn
%O A095981 0,4
%A A095981 David Callan (callan(AT)stat.wisc.edu), Jul 16 2004