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%I A190163 #6 Jul 22 2022 11:55:32
%S A190163 0,0,0,0,0,0,0,1,5,18,58,174,500,1399,3843,10421,27997,74699,198267,
%T A190163 524135,1381261,3631068,9526568,24954538,65283648,170610003,445484163,
%U A190163 1162396269,3031267533,7901082379,20586262763,53620039074,139624131310,363495081689,946147596489,2462387385085
%N 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).
%C A190163 a(n)=Sum(k*A098083(n,k), k>=0).
%F A190163 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).
%F A190163 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
%e A190163 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).
%p A190163 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);
%Y A190163 Cf. A098083, A004148
%K A190163 nonn
%O A190163 0,9
%A A190163 _Emeric Deutsch_, May 05 2011

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