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

1, 9, 108, 1620, 29160, 612360, 14696640, 396809280, 11904278400, 392841187200, 14142282739200, 551549026828800, 23165059126809600, 1042427660706432000, 50036527713908736000, 2551862913409345536000, 137800597324104658944000, 7854634047473965559808000

Offset: 0

Author

Wolfdieter Lang

Keywords

• easy
• nonn

Comments

Row m=6 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..378

Crossrefs

Cf. A032031, A007559(n+1), A034000(n+1), A034001(n+1), A051604-A051609 (rows m=0..9).

Programs

Magma

m:=30; R:=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

Maple

[seq(n!*3^(n-2)/2, n=2..18)]; # Zerinvary Lajos, Sep 23 2006
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
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

Mathematica

With[{nn = 30}, CoefficientList[Series[1/(1 - 3*x)^(9/3), {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)^(9/3))) \\ G. C. Greubel, Aug 15 2018

Formula

a(n) = ((3*n+6)(!^3))/6(!^3); e.g.f.: 1/(1-3*x)^3.

a(n) = n!*3^(n-2)/2, n >= 2. - Zerinvary Lajos, Sep 23 2006

Sum_{n>=0} 1/a(n) = 18*exp(1/3) - 24. - Amiram Eldar, Dec 18 2022