cp's OEIS Frontend

A079589 a(n) = C(5*n+1,n).

%I A079589 #45 Aug 20 2025 06:56:26
%S A079589 1,6,55,560,5985,65780,736281,8347680,95548245,1101716330,12777711870,
%T A079589 148902215280,1742058970275,20448884000160,240719591939480,
%U A079589 2840671544105280,33594090947249085,398039194165652550,4724081931321677925,56151322242892212960,668324943343021950370
%N A079589 a(n) = C(5*n+1,n).
%C A079589 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
%H A079589 Vincenzo Librandi, <a href="/A079589/b079589.txt">Table of n, a(n) for n = 0..200</a>
%F A079589 a(n) is asymptotic to c*(3125/256)^n/sqrt(n) with c=0.557.... [c = 5^(3/2)/(sqrt(Pi)*2^(7/2)) = 0.55753878629774... - _Vaclav Kotesovec_, Feb 14 2019 and Aug 20 2025]
%F A079589 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 A079589 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 A079589 a(n) = [x^n] 1/(1 - x)^(2*(2*n+1)). - _Ilya Gutkovskiy_, Oct 10 2017
%F A079589 From _Seiichi Manyama_, Aug 16 2025: (Start)
%F A079589 a(n) = Sum_{k=0..n} binomial(5*n-k,n-k).
%F A079589 G.f.: 1/(1 - x*g^3*(5+g)) where g = 1+x*g^5 is the g.f. of A002294.
%F A079589 G.f.: g^2/(5-4*g) where g = 1+x*g^5 is the g.f. of A002294.
%F A079589 G.f.: B(x)^2/(1 + 4*(B(x)-1)/5), where B(x) is the g.f. of A001449. (End)
%p A079589 seq(binomial(5*n+1,n),n=0..100); # _Robert Israel_, Aug 07 2014
%t A079589 Table[Binomial[5n+1,n],{n,0,20}]  (* _Harvey P. Dale_, Jan 23 2011 *)
%o A079589 (Magma) [Binomial(5*n+1, n): n in [0..20]]; // _Vincenzo Librandi_, Aug 07 2014
%Y A079589 Cf. A052203.
%Y A079589 Cf. A001449, A079678, A371753, A385632, A386812.
%Y A079589 Cf. A005810, A183160.
%K A079589 nonn
%O A079589 0,2
%A A079589 _Benoit Cloitre_, Jan 26 2003