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A135395 Number of walks of length 2n+3 from origin to (1,1,1) on a cubic lattice. 1

%I

%S 6,180,5040,143640,4199580,125621496,3830266440,118655943120,

%T 3724872182460,118248726796200,3789926661961440,122473276342326000,

%U 3986235855826497000,130561182081992667600,4300094066688571550400

%N Number of walks of length 2n+3 from origin to (1,1,1) on a cubic lattice.

%C a(n) is the number of walks of length 2*n+3 in a cubic lattice that begin at the origin and end at (1,1,1) using steps (1,0,0), (-1,0,0), (0,1,0), (0,-1,0), (0,0,1), (0,0,-1).

%H G. C. Greubel, <a href="/A135395/b135395.txt">Table of n, a(n) for n = 0..250</a>

%H S. Hollos and R. Hollos, <a href="http://www.exstrom.com/math/lattice/latpath.html">Lattice Paths and Walks</a>.

%F a(n) = binomial(2n+3,n) * Sum_{k=0..n} (binomial(n,k) * binomial(n+3,k+2) * binomial(2k+2,k+1)).

%F G.f.: ((12*(4*x-1)*(36*x-1)/x)*g'' + (12*(288*x^2-60*x+1)/x^2)*g' + (72*(6*x-1)/x^2)*g)/288 where g is the ogf of A002896. - _Mark van Hoeij_, Nov 12 2011

%F From _Vaclav Kotesovec_, Nov 27 2017: (Start)

%F Recurrence: n*(n+2)*(n+3)*a(n) = 4*(2*n + 3)*(5*n^2 + 10*n + 3)*a(n-1) - 36*n*(2*n + 1)*(2*n + 3)*a(n-2).

%F a(n) ~ 2^(2*n + 1) * 3^(2*n + 9/2) / (Pi*n)^(3/2). (End)

%p sq := (1-40*x+144*x^2)^(1/2); pb := 54*x*(108*x^2-27*x+1+(9*x-1)*sq);

%p H1 := hypergeom([7/6,1/3],[1],pb); H2 := hypergeom([1/6,4/3],[1],pb);

%p fa := (10-72*x-6*sq)^(1/2)/(432*x^3);

%p ogf := fa*((648*x^2-162*x+(54*x+3)*sq+5)*H1^2 - (648*x^2-342*x+(54*x+6)*sq+10)*H1*H2 - (180*x-5-3*sq)*H2^2);

%p series(ogf,x=0,20) # _Mark van Hoeij_, Nov 12 2011

%t Table[Binomial[2n+3,n]Sum[Binomial[n,k]Binomial[n+3,k+2]Binomial[2k+2,k+1],{k,0,n}],{n,0,20}] (* _Harvey P. Dale_, Mar 20 2012 *)

%o Maxima: a(n) = binomial(2n+3,n) * sum( binomial(n,k) * binomial(n+3,k+2) * binomial(2k+2,k+1), k, 0, n )

%o (PARI) a(n) = binomial(2*n+3,n) * sum(k=0,n, binomial(n,k) * binomial(n+3,k+2) * binomial(2*k+2,k+1)) \\ _Charles R Greathouse IV_, Oct 12 2016

%Y Cf. A002896.

%K easy,nonn

%O 0,1

%A Stefan Hollos (stefan(AT)exstrom.com), Dec 11 2007

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Last modified October 23 07:11 EDT 2019. Contains 328336 sequences. (Running on oeis4.)