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%I #6 Jul 22 2022 11:49:31
%S 1,1,1,2,4,8,15,28,53,102,199,391,773,1537,3075,6189,12525,25473,
%T 52037,106737,219761,454041,941089,1956357,4078010,8522016,17850512,
%U 37471531,78818748,166102378,350660371,741503529,1570402564,3330730115,7073941610,15043298781
%N Number of peakless Motzkin paths of length n containing no subwords of type uh^ju or dh^jd (j>=1), where u=(1,1), h=(1,0), and d=(1,-1) (can be easily expressed using RNA secondary structure terminology).
%C a(n)=A097100(n,0). F% G.f.: G=G(z) satisfies the equation G=1+zG+z^2*G[z+(1-z)^2*(G-zG-1)]/(1-z).
%F Conjecture D-finite with recurrence (n+2)*a(n) +(-5*n-6)*a(n-1) +2*(4*n+3)*a(n-2) +(-2*n-9)*a(n-3) +(-8*n+25)*a(n-4) +6*(n-2)*a(n-
%F 5) +9*(n-7)*a(n-6) +3*(-5*n+32)*a(n-7) +7*(n-7)*a(n-8) +(-n+8)*a(n-9)=0. - _R. J. Mathar_, Jul 22 2022
%e a(6)=15 because among the 17 (=A004148(6)) peakless Motzkin paths of length 6 only (uhu)hdd and uuh(dhd) have subwords of the forbidden type (shown between parentheses).
%p eq := G = 1+z*G+z^2*G*(z+(1-z)^2*(G-z*G-1))/(1-z): G := RootOf(eq, G): Gser := series(G, z = 0, 38): seq(coeff(Gser, z, n), n = 0 .. 35);
%Y Cf. A097100, A004148
%K nonn
%O 0,4
%A _Emeric Deutsch_, May 05 2011
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