cp's OEIS Frontend

A097779 Number of Motzkin paths of length n, starting with an up step, ending with a down step and having no peaks (can be easily expressed using RNA secondary structure terminology).

%I A097779 #8 Jul 26 2022 12:39:50
%S A097779 1,0,0,1,1,2,5,11,25,58,135,317,750,1785,4272,10275,24823,60210,
%T A097779 146576,358010,877087,2154751,5307166,13102511,32418806,80375267,
%U A097779 199650310,496803811,1238276667,3091173482,7727893389,19346109435,48493869237
%N A097779 Number of Motzkin paths of length n, starting with an up step, ending with a down step and having no peaks (can be easily expressed using RNA secondary structure terminology).
%F A097779 G.f. = z + (1-z)^2*[1-z+z^2-sqrt(1-2z-z^2-2z^3+z^4)]/(2z^2)
%F A097779 D-finite with recurrence (n+2)*a(n) -3*n*a(n-1) +(n-4)*a(n-2) +(-n+1)*a(n-3) +3*(n-5)*a(n-4) +(-n+7)*a(n-5)=0. - _R. J. Mathar_, Jul 26 2022
%e A097779 a(6)=5 because we have UHHHHD, UHDUHD, UUHHDD, UHUHDD and UUHDHD, where U=(1,1), D=(1,-1) and H=(1,0).
%p A097779 G:=z+1/2*(1-z)^2/z^2*(1-z+z^2-sqrt(1-2*z-z^2-2*z^3+z^4)): Gser:=series(G,z=0,40): 1,seq(coeff(Gser,z^n),n=1..37);
%t A097779 CoefficientList[Series[x+(1-x)^2 (1-x+x^2-Sqrt[1-2x-x^2-2x^3+x^4])/(2x^2),{x,0,40}],x] (* _Harvey P. Dale_, Dec 24 2016 *)
%Y A097779 Cf. A004148.
%K A097779 nonn
%O A097779 0,6
%A A097779 _Emeric Deutsch_, Sep 11 2004