|
|
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
|
|
|
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.: 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). - Sergey Perepechko, Jan 31 2011
|
|
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
|
|
|
KEYWORD
|
easy,nonn
|
|
AUTHOR
|
Stefan Hollos (stefan(AT)exstrom.com), Dec 11 2007
|
|
STATUS
|
approved
|
|
|
|