OFFSET
0,2
COMMENTS
a(n) is the number of paths from (0,0) to (5n,n) taking north and east steps while avoiding exactly 2 consecutive north steps. - Shanzhen Gao, Apr 15 2010
REFERENCES
Shanzhen Gao, Pattern Avoidance in Paths and Walks, in preparation [From Shanzhen Gao, Apr 15 2010]
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..200
FORMULA
a(n) is asymptotic to c*sqrt(n)*(3125/256)^n with c=0.557.... [c = 5^(3/2)/(sqrt(Pi)*2^(7/2)) = 0.55753878629774... - Vaclav Kotesovec, Feb 14 2019]
8*n*(4*n+1)*(2*n-1)*(4*n-1)*a(n) -5*(5*n+1)*(5*n-3)*(5*n-2)*(5*n-1)*a(n-1)=0. - R. J. Mathar, Jul 17 2014
G.f.: hypergeom([2/5, 3/5, 4/5, 6/5], [1/2, 3/4, 5/4], (3125/256)*x). - Robert Israel, Aug 07 2014
a(n) = [x^n] 1/(1 - x)^(2*(2*n+1)). - Ilya Gutkovskiy, Oct 10 2017
MAPLE
seq(binomial(5*n+1, n), n=0..100); # Robert Israel, Aug 07 2014
MATHEMATICA
Table[Binomial[5n+1, n], {n, 0, 20}] (* Harvey P. Dale, Jan 23 2011 *)
PROG
(Magma) [Binomial(5*n+1, n): n in [0..20]]; // Vincenzo Librandi, Aug 07 2014
CROSSREFS
KEYWORD
nonn
AUTHOR
Benoit Cloitre, Jan 26 2003
STATUS
approved