A171847 Total area under all peakless Motzkin paths of length n (n>=0).

0, 0, 0, 2, 7, 22, 68, 198, 563, 1578, 4367, 11980, 32648, 88500, 238886, 642598, 1723629, 4612170, 12316357, 32832302, 87390763, 232305470, 616812557, 1636084020, 4335770052, 11480937084, 30379110906, 80332372838, 212300488377

Offset

0,4

Links

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

Formula

a(n) = Sum_{k>=0} k*A171846(n,k).

G.f.: z^2*(g^2 -1)/((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+2)*a(n) +5*(-n-1)*a(n-1) +(7*n-4)*a(n-2) +2*(-3*n+7)*a(n-3) +(13*n-18)*a(n-4) +11*(-n+3)*a(n-5) +(13*n-60)*a(n-6) +2*(-3*n+11)*a(n-7) +(7*n-38)*a(n-8) +5*(-n+7)*a(n-9) +(n-8)*a(n-10)=0. - R. J. Mathar, Jul 22 2022

Example

a(4)=7 because the areas under the paths HHHH, HUHD, UHHD and UHDH are 0, 2, 3 and 2, respectively.

Maple

g := ((1-z+z^2-sqrt(1-2*z-z^2-2*z^3+z^4))*1/2)/z^2: G := z^2*(g^2-1)/((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, A171846.

Sequence in context: A071684 A290917 A060816 * A037552 A308113 A291012

Adjacent sequences: A171844 A171845 A171846 * A171848 A171849 A171850

Keyword

nonn

Author

Emeric Deutsch, Feb 08 2010

Status

approved