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%I #5 Nov 23 2014 16:05:49
%S 0,0,1,2,5,11,26,61,147,357,879,2183,5471,13811,35100,89724,230562,
%T 595237,1543191,4016038,10487553,27473602,72178312,190127740,
%U 502044221,1328667241,3523684572,9363119781,24924679832,66461841934,177501561659
%N Number of plateau-free Motzkin paths of length n.
%C 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 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.
%F 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.
%F Conjecture: (n+2)*a(n) +(-5*n-4)*a(n-1) +6*n*a(n-2) +(4*n-13)*a(n-3) +(-13*n+43)*a(n-4) +3*(3*n-13)*a(n-5) +(-n+4)*a(n-6) +(-n+7)*a(n-7)=0. - _R. J. Mathar_, Nov 23 2014
%e The middle two steps of UFFD form a plateau and a(4) counts the 5 paths FFUD,FUDF,UDFF,UDUD,UUDD.
%t 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}]
%K nonn
%O 0,4
%A _David Callan_, Jul 16 2004
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