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A135390 Number of walks from origin to (1,0,0) in a cubic lattice. 0
1, 15, 310, 7455, 195426, 5416026, 156061620, 4628393055, 140348412490, 4331544836190, 135614951248140, 4296741195214650, 137507314754659500, 4438467396322843500, 144329729055650881560, 4723733064176346346335 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

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

Table of n, a(n) for n=0..15.

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

FORMULA

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

Ordinary generating function - Sergey Perepechko Jan 31 2011

G(z)=1/6 * (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)

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

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 *)

PROG

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

CROSSREFS

Cf. A002896.

Sequence in context: A049381 A051691 A247238 * A105491 A158533 A284070

Adjacent sequences:  A135387 A135388 A135389 * A135391 A135392 A135393

KEYWORD

easy,nonn

AUTHOR

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

STATUS

approved

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Last modified November 17 06:06 EST 2019. Contains 329217 sequences. (Running on oeis4.)