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A094893 Total area below the lattice paths of length n defined by the rule [(0),(k)->(k-1)(k+1)] (Dyck paths). 2
1, 4, 12, 34, 84, 212, 488, 1162, 2580, 5932, 12888, 28948, 61992, 136936, 290256, 633178, 1331892, 2877308, 6016760, 12897340, 26843256, 57175384, 118545072, 251163204, 519103624, 1094915512, 2256939888, 4742198632, 9752832720 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

LINKS

G. C. Greubel, Table of n, a(n) for n = 1..1000

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+x-sqrt(1-4*x^2))/((1-2*x)*(1-4*x^2)).

a(n) ~ 3*n*2^(n-2) * (1-4*sqrt(2)/(3*sqrt(Pi*n))). - Vaclav Kotesovec, Mar 20 2014

D-finite with recurrence: n*(3*n-5)*a(n) +4*(-3*n+4)*a(n-1) +4*(-6*n^2+13*n-1)*a(n-2) +8*(6*n-5)*a(n-3) +16*(3*n-2)*(n-2)*a(n-4)=0. - R. J. Mathar, Aug 21 2018

D-finite with recurrence: n*a(n) -2*n*a(n-1) +4*(-2*n+3)*a(n-2) +8*(2*n-3)*a(n-3) +16*(n-3)*a(n-4) +32*(-n+3)*a(n-5)=0. - R. J. Mathar, Aug 21 2018

MATHEMATICA

CoefficientList[ Series[(1 + x - Sqrt[1 - 4*x^2])/((1 - 2*x)*(1 - 4*x^2)), {x, 0, 30}], x] (* Robert G. Wilson v, Jun 15 2004 *)

PROG

(PARI) x='x+O('x^50); Vec((1+x-sqrt(1-4*x^2))/((1-2*x)*(1-4*x^2))) \\ G. C. Greubel, Feb 16 2017

CROSSREFS

Sequence in context: A331834 A135373 A209818 * A036880 A107069 A191823

Adjacent sequences:  A094890 A094891 A094892 * A094894 A094895 A094896

KEYWORD

nonn,easy

AUTHOR

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

EXTENSIONS

More terms from Robert G. Wilson v, Jun 16 2004

STATUS

approved

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Last modified August 8 19:29 EDT 2020. Contains 336298 sequences. (Running on oeis4.)