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Total area under all the level steps in all peakless Motzkin paths of length n (n>=0).
1

%I #11 Jul 22 2022 12:39:13

%S 0,0,0,1,4,12,36,104,292,810,2224,6058,16408,44240,118848,318339,

%T 850608,2268206,6037892,16048945,42604344,112974302,299284044,

%U 792164740,2095161996,5537651796,14627504340,38616930931,101899265656

%N Total area under all the level steps in 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*A171848(n,k).

%F G.f.: z^3*g^2/((1+z+z^2)(1-3z+z^2)), where g=g(z) satisfies g = 1 + zg + z^2*g(g - 1).

%F Conjecture D-finite with recurrence (n+1)*a(n) -5*n*a(n-1) +(7*n-11)*a(n-2) +2*(-3*n+10)*a(n-3) +(13*n-31)*a(n-4) +11*(-n+4)*a(n-5) +(13*n-73)*a(n-6) +2*(-3*n+14)*a(n-7) +(7*n-45)*a(n-8) +5*(-n+8)*a(n-9) +(n-9)*a(n-10)=0. - _R. J. Mathar_, Jul 22 2022

%e a(4)=4 because the areas under the level steps of the paths HHHH, HUHD, UHHD, UHDH are 0, 1, 2, 1, respectively.

%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+z+z^2)*(1-3*z+z^2)): Gser := series(G, z = 0, 35): seq(coeff(Gser, z, n), n = 0 .. 30);

%Y Cf. A004148, A171848.

%K nonn

%O 0,5

%A _Emeric Deutsch_, Feb 08 2010