A094059 Analog of A054474 for walks on a 3-dimensional grid.

1, 8, 152, 5056, 205720, 9305152, 449404224, 22695553536, 1183891745688, 63293536425280, 3449750940624064, 190972642327080448, 10708174630547469632, 606900724292865506816, 34711902088494315507200, 2000990161185766676951040, 116137589109102380308573080

Offset: 0

Author

Matthijs Coster, Apr 29 2004

Keywords

• nonn

Comments

a(n) is the number of lattice paths on the 3 dimensional grid (using steps(1,1,1),(1,1,-1),(1,-1,1),(1,-1,-1),(-1,1,1),(-1,1,-1),(-1,-1,1)(-1,-1,-1)) that start and end at the origin after 2n steps, not touching the origin at intermediate stages. - Geoffrey Critzer, Feb 05 2012

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..200
• P. Flajolet and R. Sedgewick, Analytic Combinatorics, Cambridge Univ. Press, 2009, page 90.

Crossrefs

Cf. A049037.

Programs

Maple

series(2-1/hypergeom([1/4, 1/4],[1],64*x)^2, x=0, 20);  # Mark van Hoeij, Apr 16 2013

Mathematica

nn=40; a=Sum[Binomial[2n,n]^3 z^(2n), {n,0,nn}]; Select[CoefficientList[Series[2-1/a, {z,0,nn}], z], #>0&] (* Geoffrey Critzer, Feb 05 2012 *)

Formula

G.f.: 2-1/G(x) where G(x) = Sum_{n>=0} C(2n,n)^3 x^(2n). - Geoffrey Critzer, Feb 05 2012

a(n) ~ c * 64^n / n^(3/2), where c = 16*Pi^(9/2) / Gamma(1/4)^8 = 0.09252216985965964001991419323555310924034459466... . - Vaclav Kotesovec, Sep 05 2014, updated Mar 17 2024

Extensions

a(6)-a(15) added by Geoffrey Critzer, Feb 05 2012