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
KEYWORD
nonn
AUTHOR
Karol A. Penson, Sep 18 2001
EXTENSIONS
a(11) from Harry J. Smith, Sep 12 2009
STATUS
approved