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A135395 Number of walks of length 2n+3 from origin to (1,1,1) on a cubic lattice. 1
6, 180, 5040, 143640, 4199580, 125621496, 3830266440, 118655943120, 3724872182460, 118248726796200, 3789926661961440, 122473276342326000, 3986235855826497000, 130561182081992667600, 4300094066688571550400 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,1

COMMENTS

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).

LINKS

G. C. Greubel, Table of n, a(n) for n = 0..250

S. Hollos and R. Hollos, Lattice Paths and Walks.

FORMULA

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

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

From Vaclav Kotesovec, Nov 27 2017: (Start)

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).

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

MAPLE

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

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

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

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);

series(ogf, x=0, 20) # Mark van Hoeij, Nov 12 2011

MATHEMATICA

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 *)

PROG

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

(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

CROSSREFS

Cf. A002896.

Sequence in context: A253779 A046989 A210358 * A141121 A176730 A225776

Adjacent sequences:  A135392 A135393 A135394 * A135396 A135397 A135398

KEYWORD

easy,nonn

AUTHOR

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

STATUS

approved

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Last modified October 15 04:33 EDT 2019. Contains 328026 sequences. (Running on oeis4.)