A094893 Total area below the lattice paths of length n defined by the rule [(0),(k)->(k-1)(k+1)] (Dyck paths).

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

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: A135373 A338695 A209818 * A036880 A349973 A107069

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