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%I #28 Sep 08 2022 08:45:08
%S 1,6,55,560,5985,65780,736281,8347680,95548245,1101716330,12777711870,
%T 148902215280,1742058970275,20448884000160,240719591939480,
%U 2840671544105280,33594090947249085,398039194165652550
%N a(n) = C(5n+1,n).
%C 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
%D Shanzhen Gao, Pattern Avoidance in Paths and Walks, in preparation [From _Shanzhen Gao_, Apr 15 2010]
%H Vincenzo Librandi, <a href="/A079589/b079589.txt">Table of n, a(n) for n = 0..200</a>
%F 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]
%F 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
%F 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
%F a(n) = [x^n] 1/(1 - x)^(2*(2*n+1)). - _Ilya Gutkovskiy_, Oct 10 2017
%p seq(binomial(5*n+1,n),n=0..100); # _Robert Israel_, Aug 07 2014
%t Table[Binomial[5n+1,n],{n,0,20}] (* _Harvey P. Dale_, Jan 23 2011 *)
%o (Magma) [Binomial(5*n+1, n): n in [0..20]]; // _Vincenzo Librandi_, Aug 07 2014
%Y Cf. A052203.
%K nonn
%O 0,2
%A _Benoit Cloitre_, Jan 26 2003
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