A051620 - OEIS

%I #22 Sep 08 2022 08:44:59

%S 1,12,192,3840,92160,2580480,82575360,2972712960,118908518400,

%T 5231974809600,251134790860800,13059009124761600,731304510986649600,

%U 43878270659198976000,2808209322188734464000,190958233908833943552000

%N a(n) = (4*n+8)(!^4)/8(!^4), related to A034177(n+1) ((4*n+4)(!^4) quartic, or 4-factorials).

%C Row m=8 of the array A(5; m,n) := ((4*n+m)(!^4))/m(!^4), m >= 0, n >= 0.

%H G. C. Greubel, <a href="/A051620/b051620.txt">Table of n, a(n) for n = 0..363</a>

%F a(n) = ((4*n+8)(!^4))/8(!^4) = A034177(n+2)/8.

%F E.g.f.: 1/(1-4*x)^3.

%F G.f.: G(0)/2, where G(k)= 1 + 1/(1 - 2*x/(2*x + 1/(2*k+6)/G(k+1))); (continued fraction). - _Sergei N. Gladkovskii_, Jun 02 2013

%p G(x):=(1-4*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..15); # _Zerinvary Lajos_, Apr 04 2009

%t s=1;lst={s};Do[s+=n*s;AppendTo[lst, s], {n, 11, 5!, 4}];lst (* _Vladimir Joseph Stephan Orlovsky_, Nov 08 2008 *)

%t With[{nn=20},CoefficientList[Series[1/(1-4*x)^3,{x,0,nn}],x] Range[0,nn]!] (* _Harvey P. Dale_, Mar 10 2017 *)

%o (PARI) x='x+O('x^30); Vec(serlaplace(1/(1-4*x)^(12/4))) \\ _G. C. Greubel_, Aug 15 2018

%o (Magma) m:=30; R<x>:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!(1/(1-4*x)^(12/4))); [Factorial(n-1)*b[n]: n in [1..m]]; // _G. C. Greubel_, Aug 15 2018

%Y Cf. A047053, A007696(n+1), A000407, A034176(n+1), A034177(n+1), A051617-A051622 (rows m=0..10).

%K easy,nonn

%O 0,2

%A _Wolfdieter Lang_