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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
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 (list; graph; refs; listen; history; text; internal format)
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

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Last modified August 13 22:57 EDT 2020. Contains 336473 sequences. (Running on oeis4.)