A117671 a(n) = binomial(3*n+1, n+1).

1, 6, 35, 210, 1287, 8008, 50388, 319770, 2042975, 13123110, 84672315, 548354040, 3562467300, 23206929840, 151532656696, 991493848554, 6499270398159, 42671977361650, 280576272201225

Offset: 0

Author

Zerinvary Lajos, Apr 12 2006

Keywords

• nonn
• easy

Comments

a(n) = A258993(2*n+1, n). - Reinhard Zumkeller, Jun 22 2015

Examples

			if n=0 then C(3*0+1,0+1) = C(1,1) = 1.
if n=10 then C(3*10+1,10+1) = C(31,11) = 84672315.
		

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 0..1000
• Milan Janjic, Two Enumerative Functions

Crossrefs

Cf. A025174: binomial(3n-1,n-1), A006013.

Cf. A258993, A001764, A004319, A005809, A013698, A045721, A165817, A236194.

Programs

Haskell

a117671 n = a258993 (2 * n + 1) n  -- Reinhard Zumkeller, Jun 22 2015

Maple

seq(binomial(3*n+1,n+1),n=0..30); # Robert Israel, Oct 10 2017

Mathematica

Table[Binomial[3n+1,n+1],{n,0,20}] (* Harvey P. Dale, Jul 19 2011 *)

PARI

vector(30, n, n--; binomial(3*n+1, n+1)) \\ Altug Alkan, Nov 04 2015

Formula

G.f.: (2*(-1+Hypergeometric2F1[-(1/3),1/3,-(1/2),(27*x)/4]))/(3*x). - Harvey P. Dale, Jul 19 2011

G.f.: A(x) = B'(x)/B(x)-B'(x)-1/x, where B(x) = 4/3*sin(1/3*asin(sqrt((27*x)/4)))^2. - Vladimir Kruchinin, Nov 26 2014

From Peter Bala, Nov 04 2015: (Start)

With an extra initial term equal to 1, the o.g.f. equals f(x)/g(x)^2, where f(x) is the o.g.f. for A005809 and g(x) is the o.g.f. for A001764.

More generally, f(x)*g(x)^k is the o.g.f. for the sequence binomial(3*n + k,n). Cf. A045721 (k = 1), A025174 (k = 2), A004319 (k = 3), A236194 (k = 4), A013698 (k = 5), A165817 (k = -1). (End)

a(n) = [x^(2*n)] 1/(1 - x)^(n+2). - Ilya Gutkovskiy, Oct 10 2017

a(n+1) = 3*(3*n+2)*(3*n+4)*a(n)/(2*(n+2)*(2*n+1)). - Robert Israel, Oct 10 2017