A187365 Trisection of A000984 (central binomial coefficients): binomial(2(3n+2),3n+2)/3!, n>=0.

1, 42, 2145, 117572, 6686100, 388934370, 22974421470, 1372238454600, 82653088824684, 5011211083256840, 305437356823765089, 18697712969443807572, 1148770108115543559100, 70797430141465286938140, 4374750896947475198160300, 270950190057528375091435920

Offset

0,2

Comments

See a comment under A187357 concerning trisection.

This appears also in the trisection of A001700: binomial(2*(3*n+1)+1,(3*n+1)+1)/3.

Links

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

Formula

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

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

O.g.f.: (cb(x^(1/3)) - sqrt(2)*P(x^(1/3))*sqrt(1/P(x^(1/3))-(1-4*x^(1/3))/2))/(18*x^(2/3)), 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) = (1/6)*Sum_{k = 0..3*n+2} binomial(3*n+2,k)^2.

a(n) = (1/6)*hypergeom([-2 - 3*n, -2 - 3*n], [1], 1).

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

a(n) ~ 8^(2*n+1) / (3*sqrt(3*Pi*n)). - Amiram Eldar, Sep 20 2025

Mathematica

a[n_] := Binomial[3*(2*n + 1), 3*n + 2]/3; Array[a, 20, 0] (* Amiram Eldar, Sep 20 2025 *)

Crossrefs

Cf. A066802 (binomial(6n,3n)), A187364 (binomial(2*(3n+1),3n+1)/2), A000984, A002458, A100033, A187357.

Sequence in context: A382874 A216703 A258393 * A294978 A180371 A046198

Adjacent sequences: A187362 A187363 A187364 * A187366 A187367 A187368

Keyword

nonn,easy

Author

Wolfdieter Lang, Mar 10 2011

Status

approved