A378777 a(n) = n^2 * binomial(3*n, n).

0, 3, 60, 756, 7920, 75075, 668304, 5697720, 47070144, 379632825, 3004501500, 23417943120, 180241588800, 1372689900036, 10360604899680, 77595170756400, 577241321893632, 4268838966063525, 31404136939468020, 229951212925133700, 1676737802322198000, 12180171012442098435

Offset: 0

Author

Amiram Eldar, Dec 07 2024

Keywords

• nonn
• easy

Links

• Amiram Eldar, Table of n, a(n) for n = 0..500
• Necdet Batir, On the series Sum_{k=1..oo} binomial(3k,k)^{-1} k^{-n} x^k, Proc. Indian Acad. Sci. (Math. Sci.), Vol. 115, No. 4 (2005), pp. 371-381; arXiv preprint, arXiv:math/0512310 [math.AC], 2005.

Crossrefs

Cf. A005809, A090763, A144485.

Programs

Mathematica

a[n_] := n^2 * Binomial[3*n, n]; Array[a, 25, 0]

PARI

a(n) = n^2 * binomial(3*n, n);

Formula

a(n) = n^2 * A005809(n).

a(n) = 3 * n * A090763(n-1) = 3 * n * A144485(n) / 2.

Sum_{n>=1} 1/a(n) = 6 * arctan(sqrt(3)/(2*phi-1))^2 - log((phi^3+1)/(phi+1)^3)^2/2, where phi = ((25+3*sqrt(69))/2)^(1/3) (Batir, 2005, p. 378, eq. (3.1)).