A094891 Total area below the lattice paths of a given length defined by the rule [(0),(k)->(k-1)(k)(k+1)] (Motzkin paths).

1, 7, 34, 144, 563, 2095, 7532, 26410, 90853, 307893, 1030886, 3417450, 11235151, 36676453, 119003432, 384098710, 1234016321, 3948461521, 12588083810, 40001960362, 126745795259, 400532044957, 1262690290868, 3971944688584, 12469123686533, 39071957204695, 122222999571622

Offset

1,2

Links

Vincenzo Librandi, Table of n, a(n) for n = 1..200

D. Merlini, Generating functions for the area below some lattice paths, Discrete Mathematics and Theoretical Computer Science AC, 2003, 217-228.

Formula

G.f.: (1-sqrt(1-2*t-3*t^2))/((1-3*t)^2*(1+t)).

D-finite with recurrence: n*(2*n-3)*a(n) = 2*(4*n^2 - 3*n-3)*a(n-1) + 2*(2*n^2 - 12*n +3) *a(n-2) - 6*(n-1)*(4*n-1)*a(n-3) - 9*(n-2)*(2*n-1)*a(n-4). - Vaclav Kotesovec, Oct 24 2012

a(n) ~ 3^(n+1)*n/4*(1-4/(sqrt(3*Pi*n))). - Vaclav Kotesovec, Oct 24 2012

Maple

G:=(1-sqrt(1-2*t-3*t^2))/((1-3*t)^2*(1+t)): Gser:=series(G, t=0, 30): seq(coeff(Gser, t^n), n=1..28); # Emeric Deutsch, Dec 16 2004

Mathematica

Rest[CoefficientList[Series[(1-Sqrt[1-2x-3x^2])/((1-3x)^2 (1+x)), {x, 0, 30}], x]] (* Harvey P. Dale, Oct 20 2011 *)

Prog

(PARI) x='x+O('x^66); Vec((1-sqrt(1-2*x-3*x^2))/((1-3*x)^2*(1+x))) \\ Joerg Arndt, May 11 2013

Crossrefs

Sequence in context: A273722 A005023 A094256 * A306376 A192803 A052161

Adjacent sequences: A094888 A094889 A094890 * A094892 A094893 A094894

Keyword

nonn,easy

Author

Donatella Merlini (merlini(AT)dsi.unifi.it), Jun 16 2004

Extensions

More terms from Emeric Deutsch, Dec 16 2004

Status

approved