cp's OEIS Frontend

A190162 Number of peakless Motzkin paths of length n containing no subwords of type dh^ju (j>=1), where u=(1,1), h=(1,0), and d=(1,-1) (can be easily expressed using RNA secondary structure terminology).

%I A190162 #6 Jul 22 2022 11:53:00
%S A190162 1,1,1,2,4,8,17,36,77,167,365,805,1790,4008,9033,20477,46663,106843,
%T A190162 245691,567194,1314086,3054442,7120951,16647056,39015476,91654385,
%U A190162 215780420,509033640,1203085539,2848445175,6755095119,16044373511,38162885226,90897048648
%N A190162 Number of peakless Motzkin paths of length n containing no subwords of  type dh^ju (j>=1), where u=(1,1), h=(1,0), and d=(1,-1) (can be easily expressed using RNA secondary structure terminology).
%C A190162 a(n)=A098083(n,0).
%F A190162 G.f.: G=G(z) satisfies the equation G=1+zG+z^2*(G-1)[(1-z)G+z/(1-z)].
%F A190162 D-finite with recurrence (n+2)*a(n) +5*(-n-1)*a(n-1) +2*(4*n+1)*a(n-2) +(-6*n+5)*a(n-3) +(8*n-27)*a(n-4) +2*(-7*n+31)*a(n-5) +(13*n-71)*a(n-6) +(-7*n+47)*a(n-7) +(3*n-25)*a(n-8) +(-n+9)*a(n-9)=0. - _R. J. Mathar_, Jul 22 2022
%e A190162 a(7)=36 because among the 37 (=A004148(7)) peakless Motzkin paths of length 7 only uh(dhu)hd has a subword of the forbidden type (shown between parentheses).
%p A190162 eq := G = 1+z*G+z^2*(G-1)*((1-z)*G+z/(1-z)): G := RootOf(eq,G): Gser := series(G,z=0,38): seq(coeff(Gser,z,n), n = 0 .. 33);
%Y A190162 Cf. A098083, A004148
%K A190162 nonn
%O A190162 0,4
%A A190162 _Emeric Deutsch_, May 05 2011