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A135394 Number of walks from origin to (1,1,0) on a cubic lattice. 1
2, 48, 1200, 31920, 890820, 25768512, 766053288, 23265871200, 718834982580, 22523567008800, 714044153702880, 22861678250567520, 738191825153055000, 24011251877148076800, 786038700362427057600, 25877760367136497398720 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,1

COMMENTS

a(n) is the number of walks of length 2n+2 on a cubic lattice that begin at the origin and end at (1,1,0) 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..100

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

FORMULA

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

6*(n+2)^2*a(n) = (n+1)*((7*n+11)*A002896(n+1)-18*(2*n+1)*A002896(n)) - Sergey Perepechko, Feb 08 2011

G.f.: 1/(12*x)*(-2*(4*x-1)*(36*x-1)*x*g'' +(-720*x^2+160*x-3)*g' + (-144*x+18)*g) where g is the ogf of A002896. - Mark van Hoeij, Nov 12 2011

a(n) ~ 3^(7/2) * 36^n / (Pi^(3/2) * n^(3/2)). - Vaclav Kotesovec, Nov 27 2017

MATHEMATICA

Table[Binomial[2*n + 2, n]*Sum[Binomial[n, k]*Binomial[n + 2, k + 1]*Binomial[2*k + 1, k], {k, 0, n}], {n, 0, 25}] (* G. C. Greubel, Oct 12 2016 *)

PROG

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

(PARI) a(n) = binomial(2*n+2, n) * sum(k=0, n, binomial(n, k) * binomial(n+2, k+1) * binomial(2*k+1, k)) \\ Charles R Greathouse IV, Oct 12 2016

CROSSREFS

Cf. A002896.

Sequence in context: A024236 A119703 A012803 * A010046 A013517 A027509

Adjacent sequences:  A135391 A135392 A135393 * A135395 A135396 A135397

KEYWORD

easy,nonn,walk

AUTHOR

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

STATUS

approved

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Last modified October 23 23:51 EDT 2019. Contains 328379 sequences. (Running on oeis4.)