A051620 a(n) = (4*n+8)(!^4)/8(!^4), related to A034177(n+1) ((4*n+4)(!^4) quartic, or 4-factorials).

1, 12, 192, 3840, 92160, 2580480, 82575360, 2972712960, 118908518400, 5231974809600, 251134790860800, 13059009124761600, 731304510986649600, 43878270659198976000, 2808209322188734464000, 190958233908833943552000

Offset: 0

Author

Wolfdieter Lang

Keywords

• easy
• nonn

Comments

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

Links

• G. C. Greubel, Table of n, a(n) for n = 0..363

Crossrefs

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

Programs

Magma

m:=30; R:=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

Maple

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

Mathematica

s=1;lst={s};Do[s+=n*s;AppendTo[lst, s], {n, 11, 5!, 4}];lst (* Vladimir Joseph Stephan Orlovsky, Nov 08 2008 *)
With[{nn=20},CoefficientList[Series[1/(1-4*x)^3,{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Mar 10 2017 *)

PARI

x='x+O('x^30); Vec(serlaplace(1/(1-4*x)^(12/4))) \\ G. C. Greubel, Aug 15 2018

Formula

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

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

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