A378778 a(n) = n^2 * 2^n * binomial(3*n, n).

0, 6, 240, 6048, 126720, 2402400, 42771456, 729308160, 12049956864, 194372006400, 3076609536000, 47959947509760, 738269547724800, 11245075661094912, 169748150676357120, 2542638555345715200, 37830087271621066752, 559525260959878348800, 8232406073859904634880, 120560661522092497305600

Offset: 0

Author

Amiram Eldar, Dec 07 2024

Keywords

• nonn
• easy

References

• Jonathan Borwein, David Bailey, and Roland Girgensohn, Experimentation in Mathematics: Computational Paths to Discovery, A K Peters, Natick, MA, 2004. See p. 26.

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. See p. 379, eq. (3.6).
• Jonathan M. Borwein and Roland Girgensohn, Evaluations of binomial series, aequationes mathematicae, Vol. 70, No. 1 (2005), pp. 25-36. See p. 32, eq. (43).

Crossrefs

Cf. A005809, A007758, A228484, A378780.

Programs

Mathematica

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

PARI

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

Formula

a(n) = A007758(n) * A005809(n).

a(n) = n^2 * A228484(n).

a(n) = n * A378780(n).

a(n) == 0 (mod 6).

Sum_{n>=1} 1/a(n) = Pi^2/24 - log(2)^2/2 (Borwein et al., 2004; Borwein and Girgensohn, 2005; Batir, 2005).