A066802 a(n) = binomial(6*n,3*n).

1, 20, 924, 48620, 2704156, 155117520, 9075135300, 538257874440, 32247603683100, 1946939425648112, 118264581564861424, 7219428434016265740, 442512540276836779204, 27217014869199032015600, 1678910486211891090247320, 103827421287553411369671120, 6435067013866298908421603100

Offset: 0

Author

Benoit Cloitre, Jan 18 2002

Keywords

• nonn
• easy

Comments

For the trisection of a sequence (here A000984) given by its real o.g.f. see a comment and a reference under A187357.

Links

• Harry J. Smith, Table of n, a(n) for n = 0..100
• R. J. Mathar, The Eggenberger-Polya urn process: Probabilities of revisited ball ratios, vixra:2502.0097 (2025) Table 1

Crossrefs

Cf. A001450, A066798, A187364 (binomial(2(3n+1),3n+1)/2), A187365 (binomial(2(3n+2),3n+2)/3!).

Programs

Magma

[Binomial(6*n, 3*n): n in [0..15]]; // G. C. Greubel, Feb 17 2020

Maple

a := n -> hypergeom([-3*n, -3*n], [1], 1):
seq(simplify(a(n)), n=0..13); # Peter Luschny, Mar 19 2018

Mathematica

Table[Binomial[6n, 3n], {n,0,13}] (* Jean-François Alcover, Jun 03 2019 *)

PARI

a(n) = { binomial(6*n, 3*n) } \\ Harry J. Smith, Mar 28 2010

Sage

[binomial(6*n, 3*n) for n in (0..15)] # G. C. Greubel, Feb 17 2020

Formula

a(n) = A000984(3*n).

a(n) = Sum_{i=0..n} Sum_{j=0..n} Sum_{k=0..n} binomial(n, i)*binomial(n, j) *binomial(n, k)*binomial(3n, i+j+k). - Benoit Cloitre, Mar 08 2005

O.g.f. (with a(0):=1): (cb(x^(1/3)) + sqrt(2)*P(x^(1/3))*sqrt(1/P(x^(1/3))+1+2*x^(1/3)))/3, with cb(x):=1/sqrt(1-4*x) (o.g.f. of A000984) and P(x):=P(-1/2,4*x) = 1/sqrt(1+4*x+16*x^2) (o.g.f. of A116091, with P(x,z) the o.g.f. of the Legendre polynomials). - Wolfdieter Lang, Mar 24 2011

D-finite with recurrence n*(3n-1)*(3n-2)*a(n) = 8*(6n-5)*(6n-1)*(2n-1)*a(n-1). - R. J. Mathar, Sep 17 2012

a(n) = GegenbauerC(3*n, -3*n, -1). - Peter Luschny, May 07 2016

a(n) = hypergeom([-3*n, -3*n], [1], 1). - Peter Luschny, Mar 19 2018

a(n) ~ 2^(6*n)/sqrt(3*Pi*n). - Vaclav Kotesovec, Jun 07 2019

From Peter Bala, Feb 16 2020: (Start)

a(m*p^k) == a(m*p^(k-1)) ( mod p^(3*k) ) for prime p >= 5 and positive integers m and k.

a(n) = [(x*y)^(3*n)] (1 + x + y)^(6*n). Cf. A001448. (End)

Conjecture: a(n) = [x^n] G(x)^(2*n), where G(x) = (1 + x)*(1 - 6*x + x^2)/(2*x) + (x^2 - 1)*sqrt(1 - 14*x + x^2)/(2*x) = 1 + 10*x + 81*x^2 + 720*x^3 + .... The algebraic function G(x) satisfies the quadratic equation x*G(x)^2 - (1 - 5*x - 5*x^2 + x^3)*G(x) + (1 + x)^4 = 0. Cf. A001450. - Peter Bala, Oct 27 2022

a(n) = Sum_{k = 0..3*n} binomial(3*n+k-1, k). - Peter Bala, Jun 04 2024

O.g.f: 3F2(1/6,1/2,5/6; 1/3,2/3 ; 64*x). - R. J. Mathar, Jan 11 2025

Extensions

Extended to a(0)=1 by M. F. Hasler, Oct 06 2014