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A034001 One third of triple factorial numbers.

%I A034001 #56 Sep 08 2022 08:44:51
%S A034001 1,6,54,648,9720,174960,3674160,88179840,2380855680,71425670400,
%T A034001 2357047123200,84853696435200,3309294160972800,138990354760857600,
%U A034001 6254565964238592000,300219166283452416000,15311177480456073216000
%N A034001 One third of triple factorial numbers.
%H A034001 Muniru A Asiru, <a href="/A034001/b034001.txt">Table of n, a(n) for n = 1..100</a>
%H A034001 INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=495">Encyclopedia of Combinatorial Structures 495</a>.
%H A034001 Norihiro Nakashima and Shuhei Tsujie, <a href="https://arxiv.org/abs/1904.09748">Enumeration of Flats of the Extended Catalan and Shi Arrangements with Species</a>, arXiv:1904.09748 [math.CO], 2019.
%H A034001 N. J. A. Sloane and Thomas Wieder, <a href="https://doi.org/10.1007/s11083-004-9460-9">The Number of Hierarchical Orderings</a>, Order, Vol. 21 (2004), pp. 83-89; <a href="https://arxiv.org/abs/math/0307064">arXiv preprint</a>, arXiv:math/0307064 [math.CO], 2003.
%F A034001 3*a(n) = (3*n)!!! = Product_{j=1..n} 3*j = 3^n*n!.
%F A034001 E.g.f.: (-1 + 1/(1-3*x))/3.
%F A034001 E.g.f.: 1/(1-3*x)^2. - _Paul Barry_, Sep 14 2004. For offset 0. - _Wolfdieter Lang_, Apr 06 2017
%F A034001 D-finite with recurrence a(n) - 3*n*a(n-1) = 0. - _R. J. Mathar_, Dec 02 2012
%F A034001 From _Amiram Eldar_, Jan 08 2022: (Start)
%F A034001 Sum_{n>=1} 1/a(n) = 3*(exp(1/3)-1).
%F A034001 Sum_{n>=1} (-1)^(n+1)/a(n) = 3*(1-exp(-1/3)). (End)
%p A034001 G(x):=(1-3*x)^(n-3): f[0]:=G(x): for n from 1 to 29 do f[n]:=diff(f[n-1],x) od:x:=0:seq(f[n],n=0..16); # _Zerinvary Lajos_, Apr 04 2009
%t A034001 terms = 17;
%t A034001 CoefficientList[1/(1-3x)^2 + O[x]^terms, x] Range[0, terms-1]! (* _Jean-François Alcover_, Jul 28 2018 *)
%t A034001 Table[3^(n-1)*n!, {n,20}] (* _G. C. Greubel_, Aug 15 2019 *)
%o A034001 (GAP) List([1..20],n->3^(n-1)*Factorial(n)); # _Muniru A Asiru_, Jul 28 2018
%o A034001 (PARI) vector(20, n, 3^(n-1)*n!) \\ _G. C. Greubel_, Aug 15 2019
%o A034001 (Magma) [3^(n-1)*Factorial(n): n in [1..20]]; // _G. C. Greubel_, Aug 15 2019
%o A034001 (Sage) [3^(n-1)*factorial(n) for n in (1..20)] # _G. C. Greubel_, Aug 15 2019
%Y A034001 Cf. A007559, A034000.
%K A034001 easy,nonn
%O A034001 1,2
%A A034001 _Wolfdieter Lang_