A098057 Number of peakless Motzkin paths with no U H^j U, no D H^j D and no D H^jU (j>0), where U=(1,1), D=(1,-1) and H=(1,0) (can be easily expressed using RNA secondary structure terminology).

1, 1, 1, 2, 4, 8, 15, 27, 48, 84, 147, 257, 451, 796, 1413, 2526, 4544, 8226, 14978, 27417, 50434, 93183, 172865, 321857, 601263, 1126644, 2116968, 3987960, 7530200, 14249649, 27019301, 51327965, 97676156, 186177568, 355406479, 679425009

Offset

0,4

Links

Table of n, a(n) for n=0..35.

I. L. Hofacker, P. Schuster and P. F. Stadler, Combinatorics of RNA secondary structures, Discrete Appl. Math., 88, 1998, 207-237.

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

M. Vauchassade de Chaumont and G. Viennot, Polynômes orthogonaux et problèmes d'énumération en biologie moléculaire, Sem. Loth. Comb. B08l (1984) 79-86.

Formula

G.f.: [1-z+z^2-4z^3+2z^4-sqrt(1-2z-z^2+2z^3+z^4-4z^5+4z^6)]/[2z^2*(1-z)^3].

D-finite with recurrence (n+2)*a(n) +(-5*n-7)*a(n-1) +2*(4*n+5)*a(n-2) +(-2*n-13)*a(n-3) +3*(-2*n+5)*a(n-4) +18*(1)*a(n-5) +(17*n-101)*a(n-6) +(-25*n+154)*a(n-7) +2*(8*n-53)*a(n-8) +4*(-n+7)*a(n-9)=0. - R. J. Mathar, Jul 26 2022

Example

a(4)=4 because we have HHHH, UHDU, HUHD and UHHD; a(6)=15 because from all 17 peakless Motzkin paths of length 6 (see A004148) only (UHU)HDD and UUH(DHD) do not qualify.

Crossrefs

Cf. A004148.

Sequence in context: A344611 A182716 A143281 * A289692 A074029 A248729

Adjacent sequences: A098054 A098055 A098056 * A098058 A098059 A098060

Keyword

nonn

Author

Emeric Deutsch, Sep 11 2004

Status

approved