|
%I #18 Dec 18 2022 02:32:17
%S 1,12,180,3240,68040,1632960,44089920,1322697600,43649020800,
%T 1571364748800,61283225203200,2573895458534400,115825295634048000,
%U 5559614190434304000,283540323712149504000,15311177480456073216000,872737116385996173312000,52364226983159770398720000
%N a(n) = (3*n+9)!!!/9!!!, related to A032031 ((3*n)!!! triple factorials).
%C Row m=9 of the array A(4; m,n) := ((3*n+m)(!^3))/m(!^3), m >= 0, n >= 0.
%H G. C. Greubel, <a href="/A051609/b051609.txt">Table of n, a(n) for n = 0..377</a>
%F a(n) = ((3*n+9)(!^3))/9(!^3).
%F E.g.f.: 1/(1-3*x)^4.
%F From _Amiram Eldar_, Dec 18 2022: (Start)
%F a(n) = (n+3)!*3^(n-1)/2.
%F Sum_{n>=0} 1/a(n) = 162*exp(1/3) - 225. (End)
%p restart: G(x):=(1-3*x)^(n-5): 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..15); # _Zerinvary Lajos_, Apr 04 2009
%t With[{nn = 30}, CoefficientList[Series[1/(1 - 3*x)^(12/3), {x, 0, nn}], x]*Range[0, nn]!] (* _G. C. Greubel_, Aug 15 2018 *)
%o (PARI) x='x+O('x^30); Vec(serlaplace(1/(1-3*x)^(12/3))) \\ _G. C. Greubel_, Aug 15 2018
%o (Magma) m:=30; R<x>:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!(1/(1-3*x)^(12/3))); [Factorial(n-1)*b[n]: n in [1..m]]; // _G. C. Greubel_, Aug 15 2018
%Y Cf. A032031, A007559(n+1), A034000(n+1), A034001(n+1), A051604-A051608 (rows m=0..8).
%K easy,nonn
%O 0,2
%A _Wolfdieter Lang_
|
