A187364 Trisection of A000984 (central binomial coefficients): binomial(2(3n+1),3n+1)/2, n>=0.

1, 35, 1716, 92378, 5200300, 300540195, 17672631900, 1052049481860, 63205303218876, 3824345300380220, 232714176627630544, 14226520737620288370, 873065282167813104916, 53753604366668088230810, 3318776542511877736535400, 205397724721029574666088520

Offset: 0

Author

Wolfdieter Lang, Mar 10 2011

Keywords

• nonn
• easy

Comments

See a comment under A187363 concerning trisection.

This appears also in the trisection of A001700 (central binomials in the odd numbered Pascal rows): binomial(2*(3*n)+1,3*n+1).

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..554

Crossrefs

Cf. A066802 (binomial(6n,3n)), A187365 (binomial(2(3n+2),3n+2)/3!).

Cf. A002458, A100033.

Programs

Mathematica

Table[c=3n+1;Binomial[2c,c]/2,{n,0,20}] (* Harvey P. Dale, May 10 2012 *)

Formula

a(n) = binomial(2*(3*n+1),3*n+1)/2, n>=0.

a(n) = binomial(2*(3*n)+1,3*n+1), n>=0.

O.g.f.: (cb(x^(1/3)) - sqrt(2)*P(x^(1/3))*sqrt(1/P(x^(1/3))-(1+8*x^(1/3))/2))/(6*x^(1/3)), 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).

From Peter Bala, Mar 19 2023: (Start)

a(n) = (1/2)*Sum_{k = 0..3*n+1} binomial(3*n+1,k)^2.

a(n) = (1/2)*hypergeom([-1 - 3*n, -1 - 3*n], [1], 1).

a(n) = 8*(2*n - 1)*(6*n + 1)*(6*n - 1)/(n*(3*n + 1)*(3*n - 1)) * a(n-1). (End)

Right-hand side of the binomial sum identity (1/18) * Sum_{k = 0..6*n+3} (-1)^(n+k) * (k/(2*n + 1))^2 * binomial(6*n+3, k)^2 = a(n). - Peter Bala, Nov 05 2024