This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A064350 #34 May 02 2025 07:58:10
%S A064350 1,6,360,60480,19958400,10897286400,8892185702400,10137091700736000,
%T A064350 15388105201717248000,30006805143348633600000,
%U A064350 73096577329197271449600000,217535414131691079834009600000,776601428450137155007414272000000,3275704825202678519821273399296000000
%N A064350 a(n) = (3*n)!/n!.
%C A064350 Also a(n) = (((n)!)^2)*A006480(n). [corrected by _Johannes W. Meijer_, Mar 02 2009]
%C A064350 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
%H A064350 Harry J. Smith, <a href="/A064350/b064350.txt">Table of n, a(n) for n = 0..70</a>
%H A064350 Karol A. Penson and Allan I. Solomon, <a href="https://doi.org/10.1142/9789812777850_0066">Coherent states from combinatorial sequences</a>, in: E. Kapuscik and A. Horzela (eds.), Quantum theory and symmetries, World Scientific, 2002, pp. 527-530; <a href="https://arxiv.org/abs/quant-ph/0111151">arXiv preprint</a>, arXiv:quant-ph/0111151, 2001.
%F A064350 Integral representation as n-th moment of a positive function on a positive half-axis: a(n) = Integral_{x=0..oo} x^n*BesselK(1/3, 2*sqrt(x/27))/(3*Pi*sqrt(x)) dx, n >= 0.
%F A064350 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
%F A064350 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
%F A064350 E.g.f.: (with interpolated zeros) exp(x^3). - _Geoffrey Critzer_, Dec 30 2012
%F A064350 Sum_{n>=1} 1/a(n) = A248759. - _Amiram Eldar_, Nov 15 2020
%t A064350 Table[(3n)!/n!,{n,0,20}] (* _Geoffrey Critzer_, Dec 30 2012 *)
%o A064350 (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
%Y A064350 Cf. A006480, A248759.
%Y A064350 From _Johannes W. Meijer_, Mar 07 2009: (Start)
%Y A064350 Equals A001525*3!
%Y A064350 Equals row sums of A157704 and A157705. (End)
%K A064350 nonn,easy
%O A064350 0,2
%A A064350 _Karol A. Penson_, Sep 18 2001
%E A064350 a(11) from _Harry J. Smith_, Sep 12 2009