A135394 Number of walks of length 2n+2 from origin to (1,1,0) on a cubic lattice.

2, 48, 1200, 31920, 890820, 25768512, 766053288, 23265871200, 718834982580, 22523567008800, 714044153702880, 22861678250567520, 738191825153055000, 24011251877148076800, 786038700362427057600, 25877760367136497398720

Offset: 0

Author

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

Keywords

• easy
• nonn
• walk

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.

Crossrefs

Cf. A002896.

Programs

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 *)

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

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 o.g.f. 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