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%I #30 Mar 17 2024 06:33:15
%S 1,8,152,5056,205720,9305152,449404224,22695553536,1183891745688,
%T 63293536425280,3449750940624064,190972642327080448,
%U 10708174630547469632,606900724292865506816,34711902088494315507200,2000990161185766676951040,116137589109102380308573080
%N Analog of A054474 for walks on a 3-dimensional grid.
%C 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
%H Alois P. Heinz, <a href="/A094059/b094059.txt">Table of n, a(n) for n = 0..200</a>
%H P. Flajolet and R. Sedgewick, <a href="http://algo.inria.fr/flajolet/Publications/AnaCombi/anacombi.html">Analytic Combinatorics</a>, Cambridge Univ. Press, 2009, page 90.
%F G.f.: 2-1/G(x) where G(x) = Sum_{n>=0} C(2n,n)^3 x^(2n). - _Geoffrey Critzer_, Feb 05 2012
%F 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
%p series(2-1/hypergeom([1/4, 1/4],[1],64*x)^2, x=0, 20); # _Mark van Hoeij_, Apr 16 2013
%t 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 *)
%Y Cf. A049037.
%K nonn
%O 0,2
%A _Matthijs Coster_, Apr 29 2004
%E a(6)-a(15) added by _Geoffrey Critzer_, Feb 05 2012
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