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%I #32 Apr 18 2018 19:53:34
%S 0,1,8,50,288,1605,8824,48286,264128,1447338,7953040,43842788,
%T 242507456,1345868589,7493458392,41850173670,234408444288,
%U 1316541032958,7413214297968,41842633282620,236703844320960
%N Number of walks on simple cubic lattice (starting on the xy plane, never going below it and finishing a height 1 above it).
%H Vincenzo Librandi, <a href="/A052177/b052177.txt">Table of n, a(n) for n = 0..200</a>
%H Rigoberto Flórez, Leandro Junes, José L. Ramírez, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL21/Florez/florez4.html">Further Results on Paths in an n-Dimensional Cubic Lattice</a>, Journal of Integer Sequences, Vol. 21 (2018), Article 18.1.2.
%H R. K. Guy, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL3/GUY/catwalks.html">Catwalks, sandsteps and Pascal pyramids</a>, J. Integer Sequences, Vol. 3 (2000), Article #00.1.6.
%F a(n) = 4*a(n-1)+A005572(n-1)+A052178(n-1) = A052179(n, 1) = Sum_{j=0..ceiling((n-1)/2)} 4^(n-2j-1)*binomial(n, 2j+1)*binomial(2j+2, j+1)/(j+2).
%F Recurrence: (n-1)*(n+3)*a(n) = 4*n*(2*n+1)*a(n-1) - 12*(n-1)*n*a(n-2). - _Vaclav Kotesovec_, Oct 08 2012
%F a(n) ~ 6^(n+3/2)/(sqrt(Pi)*n^(3/2)). - _Vaclav Kotesovec_, Oct 08 2012
%F G.f.: (1 - 4*x - sqrt(1-8*x+12*x^2))^2/(4*x^3). - _Mark van Hoeij_, May 16 2013
%t Flatten[{0,RecurrenceTable[{(n-1)*(n+3)*a[n] == 4*n*(2*n+1)*a[n-1] - 12*(n-1)*n*a[n-2],a[1]==1,a[2]==8},a,{n,20}]}] (* _Vaclav Kotesovec_, Oct 08 2012 *)
%K nonn,walk
%O 0,3
%A _N. J. A. Sloane_, Jan 26 2000
%E More terms and formula from _Henry Bottomley_, Aug 23 2001
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