A190161 Number of subwords of type uh^ju and dh^jd (j>=1), where u=(1,1), h=(1,0), and d=(1,-1), in all peakless Motzkin paths of length n (can be easily expressed using RNA secondary structure terminology).

0, 0, 0, 0, 0, 0, 2, 10, 34, 104, 300, 834, 2270, 6094, 16206, 42818, 112606, 295138, 771616, 2013550, 5246954, 13657882, 35522364, 92331014, 239875614, 622971814, 1617463986, 4198716114, 10897812738, 28282859174, 73398069768, 190474295608, 494298218888, 1282776917922

Offset

0,7

Comments

a(n)=Sum(k*A097100(n,k), k>=0).

Links

Table of n, a(n) for n=0..33.

Formula

G.f.: G(z)=2z^3*g^3*(g-1)/[(1-z)(1-z^2*g^2)], where g=1+zg+z^2*g(g-1).

Example

a(6)=2 because among the 17 (=A004148(6)) peakless  Motzkin paths of length 6 only (uhu)hdd and uuh(dhd) have subwords of the prescribed type (shown between parentheses).

Maple

eq := g = 1+z*g+z^2*g*(g-1): g := RootOf(eq, g): gser := series(2*z^5*g^3*(g-1)/((1-z)*(1-z^2*g^2)), z = 0, 38): seq(coeff(gser, z, n), n = 0 .. 33);

Crossrefs

Cf. A097100, A004148

Sequence in context: A211905 A022498 A036799 * A196969 A351551 A351550

Adjacent sequences: A190158 A190159 A190160 * A190162 A190163 A190164

Keyword

nonn

Author

Emeric Deutsch, May 05 2011

Status

approved