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

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

Offset: 0

Author

Amiram Eldar, Dec 07 2024

Keywords

• nonn
• easy

Links

• 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.

Crossrefs

Cf. A005809, A128789, A378778, A378780.

Cf. A002117, A006752.

Programs

Mathematica

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

PARI

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

Formula

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).