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A095981 Number of plateau-free Motzkin paths of length n. 0
0, 0, 1, 2, 5, 11, 26, 61, 147, 357, 879, 2183, 5471, 13811, 35100, 89724, 230562, 595237, 1543191, 4016038, 10487553, 27473602, 72178312, 190127740, 502044221, 1328667241, 3523684572, 9363119781, 24924679832, 66461841934, 177501561659 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,4

COMMENTS

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.

LINKS

Table of n, a(n) for n=0..30.

FORMULA

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.

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

EXAMPLE

The middle two steps of UFFD form a plateau and a(4) counts the 5 paths FFUD,FUDF,UDFF,UDUD,UUDD.

MATHEMATICA

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}]

CROSSREFS

Sequence in context: A191692 A182015 A124217 * A247471 A082397 A051286

Adjacent sequences:  A095978 A095979 A095980 * A095982 A095983 A095984

KEYWORD

nonn

AUTHOR

David Callan, Jul 16 2004

STATUS

approved

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Last modified February 25 08:42 EST 2020. Contains 332221 sequences. (Running on oeis4.)