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

Author

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

Keywords

• nonn
• easy

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.

Programs

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 *)

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

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

Extensions

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