login
A135394
Number of walks of length 2n+2 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
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
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 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
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
KEYWORD
easy,nonn,walk
AUTHOR
Stefan Hollos (stefan(AT)exstrom.com), Dec 11 2007
STATUS
approved