|
%I #14 Jul 22 2022 11:48:09
%S 0,0,0,1,3,6,12,25,53,115,255,575,1315,3043,7111,16756,39766,94961,
%T 228003,550081,1332839,3241930,7913028,19375635,47579847,117149125,
%U 289142441,715253644,1773011502,4403539181,10956537307,27307002454,68164324150,170404155586,426584025250,1069289177950
%N Number of uh^jd's for some j>0, starting at level 0, 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 The terms a(n), starting from n=3, are the partial sums of the sequence A089735.
%F G.f.: z^3*g^2/(1-z), where g=1+z*g+z^2*g*(g-1).
%F a(n) = Sum_{k>=0} k*A098071(n,k).
%F From _Vaclav Kotesovec_, May 29 2022: (Start)
%F G.f.: (-1 + x - x^2 + sqrt((1 + (-3 + x)*x) * (1 + x + x^2)))^2 / (4*(1-x)*x).
%F a(n) ~ 5^(1/4) * phi^(2*n-1) / (sqrt(Pi) * n^(3/2)), where phi = A001622 is the golden ratio. (End)
%F D-finite with recurrence (n+1)*a(n) +(-4*n+1)*a(n-1) +(5*n-8)*a(n-2) +(-5*n+18)*a(n-3) +(5*n-22)*a(n-4) +(-5*n+32)*a(n-5) +(4*n-31)*a(n-6) +(-n+9)*a(n-7)=0. - _R. J. Mathar_, Jul 22 2022
%e a(4)=3 because the 4 (=A004148(4)) peakless Motzkin paths of length 4, namely hhhh, h(uhd), (uhd)h, and (uhhd) contain 0+1+1+1 subwords of type uh^ju for some j>0, starting at level 0 (shown between parentheses.
%p eq := g = 1+z*g+z^2*g*(g-1): g := RootOf(eq, g): F := z^3*g^2/(1-z): Fser := series(F, z = 0, 38): seq(coeff(Fser, z, n), n = 0 .. 35);
%t CoefficientList[Series[(-1 + x - x^2 + Sqrt[(1 + (-3 + x)*x)*(1 + x + x^2)])^2 / (4*(1 - x)*x), {x, 0, 40}], x] (* _Vaclav Kotesovec_, May 29 2022 *)
%Y Cf. A098071, A004148, A089735.
%K nonn
%O 0,5
%A _Emeric Deutsch_, May 05 2011
|
