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

0,2

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.

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

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 *)

Crossrefs

Cf. A049037.

Sequence in context: A360338 A059510 A264708 * A171202 A247538 A360575

Adjacent sequences: A094056 A094057 A094058 * A094060 A094061 A094062

Keyword

nonn

Author

Matthijs Coster, Apr 29 2004

Extensions

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

Status

approved