A064350 a(n) = (3*n)!/n!.

1, 6, 360, 60480, 19958400, 10897286400, 8892185702400, 10137091700736000, 15388105201717248000, 30006805143348633600000, 73096577329197271449600000, 217535414131691079834009600000

Offset

0,2

Comments

Also a(n) = (((n)!)^2)*A006480(n). [corrected by Johannes W. Meijer, Mar 02 2009]

a(n) is the number of ways to partition the set {1,2,...,3n} into n blocks of size 3 and then linearly order the elements within each block. - Geoffrey Critzer, Dec 30 2012

Links

Harry J. Smith, Table of n, a(n) for n = 0..70

Karol A. Penson and Allan I. Solomon, Coherent states from combinatorial sequences, in: E. Kapuscik and A. Horzela (eds.), Quantum theory and symmetries, World Scientific, 2002, pp. 527-530; arXiv preprint, arXiv:quant-ph/0111151, 2001.

Formula

Integral representation as n-th moment of a positive function on a positive half-axis, in Maple notation: a(n)=int(x^n*BesselK(1/3, 2*sqrt(x/27))/(3*Pi*sqrt(x)), x=0..infinity), n=0, 1, ...

A recursive formula: a(n) = (27 * (n - 1)^2 + 27 * (n - 1) + 6) * a(n - 1) with a(0) = 1. An explicit formula following from the recursion equation: a(n) = (3/2)*27^n*GAMMA(n+2/3)*GAMMA(n+1/3)/(Pi*3^(1/2)). - Thomas Wieder, Nov 15 2004

E.g.f.: (of aerated sequence) 2*cos(arcsin((3*sqrt(3)*x/2)/3))/sqrt(4-27*x^2). - Paul Barry, Jul 27 2010

E.g.f.: (with interpolated zeros) exp(x^3). - Geoffrey Critzer, Dec 30 2012

Sum_{n>=1} 1/a(n) = A248759. - Amiram Eldar, Nov 15 2020

Mathematica

Table[(3n)!/n!, {n, 0, 20}]  (* Geoffrey Critzer, Dec 30 2012 *)

Prog

(PARI) { t=f=1; for (n=0, 70, if (n, t*=3*n*(3*n - 1)*(3*n - 2); f*=n); write("b064350.txt", n, " ", t/f) ) } \\ Harry J. Smith, Sep 12 2009

Crossrefs

Cf. A006480, A248759.

From Johannes W. Meijer, Mar 07 2009: (Start)

Equals A001525*3!

Equals row sums of A157704 and A157705. (End)

Sequence in context: A036281 A202367 A262179 * A069945 A086205 A173608

Adjacent sequences: A064347 A064348 A064349 * A064351 A064352 A064353

Keyword

nonn

Author

Karol A. Penson, Sep 18 2001

Extensions

a(11) from Harry J. Smith, Sep 12 2009

Status

approved