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A051606 a(n) = (3*n+6)!!!/6!!!, related to A032031 ((3*n)!!! triple factorials).

%I A051606 #20 Dec 18 2022 02:30:39
%S A051606 1,9,108,1620,29160,612360,14696640,396809280,11904278400,
%T A051606 392841187200,14142282739200,551549026828800,23165059126809600,
%U A051606 1042427660706432000,50036527713908736000,2551862913409345536000,137800597324104658944000,7854634047473965559808000
%N A051606 a(n) = (3*n+6)!!!/6!!!, related to A032031 ((3*n)!!! triple factorials).
%C A051606 Row m=6 of the array A(4; m,n) := ((3*n+m)(!^3))/m(!^3), m >= 0, n >= 0.
%H A051606 G. C. Greubel, <a href="/A051606/b051606.txt">Table of n, a(n) for n = 0..378</a>
%F A051606 a(n) = ((3*n+6)(!^3))/6(!^3); e.g.f.: 1/(1-3*x)^3.
%F A051606 a(n) = n!*3^(n-2)/2, n >= 2. - _Zerinvary Lajos_, Sep 23 2006
%F A051606 Sum_{n>=0} 1/a(n) = 18*exp(1/3) - 24. - _Amiram Eldar_, Dec 18 2022
%p A051606 [seq(n!*3^(n-2)/2, n=2..18)]; # _Zerinvary Lajos_, Sep 23 2006
%p A051606 with(combstruct):ZL:=[T,{T=Union(Z,Prod(Epsilon,Z,T),Prod(T,Z,Epsilon),Prod(T,Z))},labeled]:seq(count(ZL,size=i)/6,i=2..18); # _Zerinvary Lajos_, Dec 16 2007
%p A051606 restart: G(x):=(1-3*x)^(n-4): 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 A051606 With[{nn = 30}, CoefficientList[Series[1/(1 - 3*x)^(9/3), {x, 0, nn}], x]*Range[0, nn]!] (* _G. C. Greubel_, Aug 15 2018 *)
%o A051606 (PARI) x='x+O('x^30); Vec(serlaplace(1/(1-3*x)^(9/3))) \\ _G. C. Greubel_, Aug 15 2018
%o A051606 (Magma) m:=30; R<x>:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!(1/(1-3*x)^(9/3))); [Factorial(n-1)*b[n]: n in [1..m]]; // _G. C. Greubel_, Aug 15 2018
%Y A051606 Cf. A032031, A007559(n+1), A034000(n+1), A034001(n+1), A051604-A051609 (rows m=0..9).
%K A051606 easy,nonn
%O A051606 0,2
%A A051606 _Wolfdieter Lang_