A378780 -id:A378780 - cp's OEIS frontend

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

Amiram Eldar, Dec 07 2024

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

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).

Cf. A005809, A007758, A228484, A378780.

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);
```

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).

A378781 a(n) = n^3 * 2^n * binomial(3*n, n) / 3.

Original entry on oeis.org

0, 2, 160, 6048, 168960, 4004000, 85542912, 1701719040, 32133218304, 583116019200, 10255365120000, 175853140869120, 2953078190899200, 48728661198077952, 792158036489666560, 12713192776728576000, 201760465448645689344, 3170643145439310643200, 49394436443159427809280

Offset: 0

Amiram Eldar, Dec 07 2024

Amiram Eldar, Table of n, a(n) for n = 0..500
D. V. Chudnovsky and G. V. Chudnovsky, Classification of hypergeometric identities for Pi and other logarithms of algebraic numbers, Proceedings of the National Academy of Sciences, Vol. 95, No. 6 (1998), pp. 2744-2749. See p. 2749.

Cf. A005809, A128789, A378778, A378780.

Cf. A002117, A006752.

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

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

a(n) = A128789(n) * A005809(n) / 3.
a(n) = n * A378778(n) / 3.
a(n) = n^2 * A378780(n) / 3.
Sum_{n>=1} 1/a(n) = 3*G*Pi - Pi^2*log(2)/8 + log(2)^3/2 - 99*zeta(3)/16, where G is Catalan's constant (Chudnovsky and Chudnovsky, 1998).

cp's OEIS Frontend

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

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