cp's OEIS Frontend

A187258 Number of UH^jD's for some j>0, in all peakless Motzkin paths of length n (here U=(1,1), D=(1,-1) and H=(1,0); can be easily expressed using RNA secondary structure terminology).

%I A187258 #20 Jan 11 2024 10:47:46
%S A187258 0,0,0,1,3,7,17,41,99,242,596,1477,3681,9215,23155,58368,147530,
%T A187258 373768,948882,2413264,6147414,15682008,40056238,102434119,262228051,
%U A187258 671945055,1723350315,4423518544,11362907022,29208834520,75131251334,193370093508,497969663062
%N A187258 Number of UH^jD's for some j>0, in all peakless Motzkin paths of length n (here U=(1,1), D=(1,-1) and H=(1,0); can be easily expressed using RNA secondary structure terminology).
%C A187258 Same as A089742 except for the first three terms. - _Georg Fischer_, Oct 14 2018
%F A187258 G.f.: z^3*G^2/((1-z)*(1-z^2*G^2)), where G = 1+z*G+z^2*G*(G-1).
%F A187258 a(n) = Sum_{k>=0} k*A089741(n,k).
%F A187258 D-finite with recurrence (-n+1)*a(n) +(4*n-7)*a(n-1) +(-5*n+16)*a(n-2) +(5*n-22)*a(n-3) +(-5*n+18)*a(n-4) +(5*n-24)*a(n-5) +(-4*n+25)*a(n-6) +(n-7)*a(n-7)=0. - _R. J. Mathar_, Jul 22 2022
%e A187258 a(4)=3 because in HHHH, HUHD, UHDH and UHHD we have 0+1+1+1 subwords of the type UH^jD.
%p A187258 eq := g = 1+z*g+z^2*g*(g-1): G := RootOf(eq, g): Gser := series(z^3*G^2/((1-z)*(1-z^2*G^2)), z = 0, 35): seq(coeff(Gser, z, n), n = 0 .. 32);
%Y A187258 Cf. A004148, A089741, A089742.
%K A187258 nonn
%O A187258 0,5
%A A187258 _Emeric Deutsch_, May 05 2011