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

1, 6, 55, 560, 5985, 65780, 736281, 8347680, 95548245, 1101716330, 12777711870, 148902215280, 1742058970275, 20448884000160, 240719591939480, 2840671544105280, 33594090947249085, 398039194165652550, 4724081931321677925, 56151322242892212960, 668324943343021950370

Offset: 0

Author

Benoit Cloitre, Jan 26 2003

Keywords

• nonn

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

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..200

Crossrefs

Cf. A052203.

Cf. A001449, A079678, A371753, A385632, A386812.

Cf. A005810, A183160.

Programs

Magma

[Binomial(5*n+1, n): n in [0..20]]; // Vincenzo Librandi, Aug 07 2014

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 *)

Formula

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]

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

From Seiichi Manyama, Aug 16 2025: (Start)

a(n) = Sum_{k=0..n} binomial(5*n-k,n-k).

G.f.: 1/(1 - x*g^3*(5+g)) where g = 1+x*g^5 is the g.f. of A002294.

G.f.: g^2/(5-4*g) where g = 1+x*g^5 is the g.f. of A002294.

G.f.: B(x)^2/(1 + 4*(B(x)-1)/5), where B(x) is the g.f. of A001449. (End)