0,5

Number of Motzkin paths with no peaks and no valleys, i.e. no UD's and no DU's, where U=(1,1) and D=(1,-1). Example: a(7)=16 because there are 17 peakless Motzkin paths of length 6 (see A004148) of which only UHDUHD has a valley (here H=(1,0)). - Emeric Deutsch (deutsch(AT)duke.poly.edu), Jan 08 2004

a(n+2) = number of Motzkin n-paths that avoid both UU and DD = number of Motzkin n-paths that avoid both UU and UFU. Example: a(7)=16 since of the 21 Motzkin 5-paths, only FUUDD, UFUDD, UUDDF, UUDFD, UUFDD contain a UU or DD (or both). Likewise, only FUUDD, UFUDD, UUDDF, UUDFD, UUFDD contain a UU or UFU. - David Callan (callan(AT)stat.wisc.edu), Jul 15 2004

The g.f. (z**3-z**2+2*z-1)*(z**3-z**2-2*z+1)/(2*z-1)/(2*z**5+z**3+z**2-3*z+1) conjectured by S. Plouffe in his 1992 dissertation is wrong.

T. Doslic, D. Svrtan and D. Veljan, Enumerative aspects of secondary structures, Discr. Math., 285 (2004), 67-82.

P. R. Stein and M. S. Waterman, On some new sequences generalizing the Catalan and Motzkin numbers, Discrete Math., 26 (1979), 261-272.

E. J. J. van Rensburg, Adsorbing bargraph paths in a q-wedge, Journal of Physics A, v.38 n.40, 8505-8525.

T. D. Noe, Table of n, a(n) for n=0..200

M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210.

S. Plouffe, Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.

S. Plouffe, 1031 Generating Functions and Conjectures, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.

M. S. Waterman, Home Page (contains copies of his papers)

G.f.=2/[1-z+z^2+z^3+sqrt[(1-z^4)(1-2z-z^2)]]. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Jan 08 2004

For Maple code see A004148.

a[ 0 ]=1; a[ n_Integer ] := a[ n ]=a[ n-1 ]+Sum[ a[ k ]*a[ n-2-k ], {k, 2, n-2} ];

(PARI) a(n)=polcoeff((1-x+x^2+x^3-sqrt((1-x^4)*(1-2*x-x^2)+x^3*O(x^n)))/(2*x^2), n) /* Michael Somos Oct 28 2005 */

Third row of A064645.

Cf. A004148, A001006.

Adjacent sequences: A004146 A004147 A004148 this_sequence A004150 A004151 A004152

Sequence in context: A098588 A126683 A005821 this_sequence A129986 A110334 A084636

nonn,nice

njas