A135390 Number of walks from origin to (1,0,0) in a cubic lattice.

1, 15, 310, 7455, 195426, 5416026, 156061620, 4628393055, 140348412490, 4331544836190, 135614951248140, 4296741195214650, 137507314754659500, 4438467396322843500, 144329729055650881560, 4723733064176346346335

Offset: 0

Author

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

Keywords

• easy
• nonn

Comments

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

Links

• Stefan Hollos and Richard Hollos, Lattice Paths and Walks.
• Nobu C. Shirai and Naoyuki Sakumichi, Universal Negative Energetic Elasticity in Polymer Chains: Crossovers among Random, Self-Avoiding, and Neighbor-Avoiding Walks, arXiv:2408.14992 [cond-mat.soft], 2024. See p. 5.

Crossrefs

Cf. A002896.

Programs

Mathematica

f[n_] := Binomial[2 n + 1, n]*Sum[ Binomial[n, k]*Binomial[n + 1, k]*Binomial[2 k, k], {k, 0, n}]; Array[f, 16, 0] (* Robert G. Wilson v *)

Maxima

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

Formula

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

G.f.: (1/(6*z)) * (1/sqrt(1+12*z)*hypergeom([1/8,3/8],[1],64/81*z*(1+sqrt(1-36*z))^2*(2+sqrt(1-36*z))^4/(1+12*z)^4)*hypergeom([1/8,3/8],[1],64/81*z*(1-sqrt(1-36*z))^2*(2-sqrt(1-36*z))^4/(1+12*z)^4) - 1). - Sergey Perepechko, Jan 31 2011

a(n) = A002896(n+1)/6.