A051609 a(n) = (3*n+9)!!!/9!!!, related to A032031 ((3*n)!!! triple factorials).

1, 12, 180, 3240, 68040, 1632960, 44089920, 1322697600, 43649020800, 1571364748800, 61283225203200, 2573895458534400, 115825295634048000, 5559614190434304000, 283540323712149504000, 15311177480456073216000, 872737116385996173312000, 52364226983159770398720000

Offset: 0

Author

Wolfdieter Lang

Keywords

• easy
• nonn

Comments

Row m=9 of the array A(4; m,n) := ((3*n+m)(!^3))/m(!^3), m >= 0, n >= 0.

Links

• G. C. Greubel, Table of n, a(n) for n = 0..377

Crossrefs

Cf. A032031, A007559(n+1), A034000(n+1), A034001(n+1), A051604-A051608 (rows m=0..8).

Programs

Magma

m:=30; R:=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

Maple

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

Mathematica

With[{nn = 30}, CoefficientList[Series[1/(1 - 3*x)^(12/3), {x, 0, nn}], x]*Range[0, nn]!] (* G. C. Greubel, Aug 15 2018 *)

PARI

x='x+O('x^30); Vec(serlaplace(1/(1-3*x)^(12/3))) \\ G. C. Greubel, Aug 15 2018

Formula

a(n) = ((3*n+9)(!^3))/9(!^3).

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

From Amiram Eldar, Dec 18 2022: (Start)

a(n) = (n+3)!*3^(n-1)/2.

Sum_{n>=0} 1/a(n) = 162*exp(1/3) - 225. (End)