A190163 Number of subwords of type dh^ju (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, 0, 1, 5, 18, 58, 174, 500, 1399, 3843, 10421, 27997, 74699, 198267, 524135, 1381261, 3631068, 9526568, 24954538, 65283648, 170610003, 445484163, 1162396269, 3031267533, 7901082379, 20586262763, 53620039074, 139624131310, 363495081689, 946147596489, 2462387385085

Offset

0,9

Comments

a(n)=Sum(k*A098083(n,k), k>=0).

Links

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

Formula

G.f.: G(z)=z^5*g^2*(g-1)^2/[(1-z)(1-z^2*g^2)], where g=1+zg+z^2*g(g-1).

Conjecture D-finite with recurrence -4*(n+1)*(n-7)*a(n) +(13*n^2-85*n+28)*a(n-1) +(-7*n^2+52*n-41)*a(n-2) +(5*n^2-41*n+67)*a(n-3) +(-13*n^2+103*n-197)*a(n-4) +(7*n-29)*(n-5)*a(n-5) -(n-5)*(n-6)*a(n-6)=0. - R. J. Mathar, Jul 22 2022

Example

a(7)=1 because among the 37 (=A004148(7)) peakless  Motzkin paths of length 7 only uh(dhu)hd  has a subword of the prescribed type (shown between parentheses).

Maple

eq := g = 1+z*g+z^2*g*(g-1): g := RootOf(eq, g): G := z^5*g^2*(g-1)^2/((1-z)*(1-z^2*g^2)): Gser := series(G, z = 0, 38): seq(coeff(Gser, z, n), n = 0 .. 35);

Crossrefs

Cf. A098083, A004148

Sequence in context: A272583 A247717 A128553 * A364553 A335551 A235612

Adjacent sequences: A190160 A190161 A190162 * A190164 A190165 A190166

Keyword

nonn

Author

Emeric Deutsch, May 05 2011

Status

approved