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
Keywords
Examples
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).
Programs
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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);
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).
Comments