A171849 Total area under all the level steps in all peakless Motzkin paths of length n (n>=0).

0, 0, 0, 1, 4, 12, 36, 104, 292, 810, 2224, 6058, 16408, 44240, 118848, 318339, 850608, 2268206, 6037892, 16048945, 42604344, 112974302, 299284044, 792164740, 2095161996, 5537651796, 14627504340, 38616930931, 101899265656

Offset

0,5

Links

Table of n, a(n) for n=0..28.

R. Willenbring, RNA structure, permutations and statistics, Discrete Appl. Math., 157, 2009, 1607-1614.

Formula

a(n) = Sum_{k>=0} k*A171848(n,k).

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).

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

Example

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

Maple

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);

Crossrefs

Cf. A004148, A171848.

Sequence in context: A237045 A291732 A291264 * A199937 A290380 A003212

Adjacent sequences: A171846 A171847 A171848 * A171850 A171851 A171852

Keyword

nonn

Author

Emeric Deutsch, Feb 08 2010

Status

approved