A378779 a(n) = n^2 * 4^n * binomial(3*n, n).

0, 12, 960, 48384, 2027520, 76876800, 2737373184, 93351444480, 3084788957184, 99518467276800, 3150448164864000, 98221972499988480, 3023952067480780800, 92119659815689519104, 2781153700681435054080, 83317180181568395673600, 2479232599432958230659072, 73338095004533174933913600

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, A006588, A128782.

Programs

Mathematica

a[n_] := n^2 * 4^n * Binomial[3*n, n]; Array[a, 25, 0]

PARI

a(n) = n^2 * 4^n * binomial(3*n, n);

Formula

a(n) = A128782(n) * A005809(n).

a(n) = n^2 * A006588(n).

a(n) == 0 (mod 12).

Sum_{n>=1} (-1)^n/a(n) = 6 * arccot(2*sqrt(3)+sqrt(7))^2 - log(2)^2/2 (Batir, 2005, p. 379, eq. (3.8)).