%I #11 Jul 22 2022 12:41:18
%S 0,0,0,1,4,14,46,140,412,1186,3354,9368,25920,71182,194322,527927,
%T 1428530,3852594,10360700,27795561,74414408,198862280,530590812,
%U 1413712094,3762056094,10000260036,26556402534,70459947925,186796151768
%N The difference between the area under a peakless Motzkin path and the number of its U-steps, summed over all peakless Motzkin paths of length n (n>=0).
%H R. Willenbring, <a href="https://doi.org/10.1016/j.dam.2008.10.002">RNA structure, permutations and statistics</a>, Discrete Appl. Math., 157, 2009, 1607-1614.
%F a(n) = Sum_{k>=0} k*A171850(n,k).
%F G.f.: z^3*g^2(1 - 2z + 2zg)/(1 - z + z^2 - 2z^2*g)^2, where g=g(z) satisfies g = 1 + zg + z^2*g(g - 1).
%F Conjecture D-finite with recurrence -2*(n+2)*(1088*n-6417)*a(n) +(10968*n^2-55789*n-59252)*a(n-1) +2*(-6704*n^2+45808*n-27647)*a(n-2) +2*(2264*n^2-18729*n+42256)*a(n-3) +2*(-11968*n^2+82555*n-88362)*a(n-4) +(24904*n^2-243375*n+511724)*a(n-5) +4*(-1088*n^2+9137*n-25680)*a(n-6) +2*(6616*n^2-65333*n+164938)*a(n-7) -2*(5616*n-25895)*(n-7)*a(n-8) +(2264*n-10805)*(n-8)*a(n-9)=0. - _R. J. Mathar_, Jul 22 2022
%e a(4)=4 because for the 4 (=A004148(4)) peakless Motzkin paths of length 4, namely, HHHH, HUHD, UHHD, UHDH, the areas under the paths are 0,2,3,2 and the number of U-steps are 0,1,1,1; now, (0-0) + (2-1) + (3-1) + (2-1) = 0 + 1 + 2 + 1 = 4.
%p g := ((1-z+z^2-sqrt(1-2*z-z^2-2*z^3+z^4))*1/2)/z^2: G := z^3*g^2*(1-2*z+2*z*g)/(1-z+z^2-2*z^2*g)^2: Gser := series(G, z = 0, 35): seq(coeff(Gser, z, n), n = 0 .. 30);
%Y Cf. A004148, A171850.
%K nonn
%O 0,5
%A _Emeric Deutsch_, Feb 08 2010