A052700 Expansion of e.g.f. x*(1-x)/(1-3*x).
0, 1, 4, 36, 432, 6480, 116640, 2449440, 58786560, 1587237120, 47617113600, 1571364748800, 56569130956800, 2206196107315200, 92660236507238400, 4169710642825728000, 200146110855634944000, 10207451653637382144000, 551202389296418635776000, 31418536189895862239232000
Offset: 0
Links
- G. C. Greubel, Table of n, a(n) for n = 0..375
- INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 650
Crossrefs
Cf. A153647.
Programs
-
Maple
spec := [S,{S=Prod(Z,Sequence(Prod(Sequence(Z),Union(Z,Z))))},labeled]: seq(combstruct[count](spec,size=n), n=0..20); -
Mathematica
Table[2*3^(n-2)*n! -2*Boole[n==0]/9 + Boole[n==1]/3, {n,0,30}] (* G. C. Greubel, May 31 2022 *) With[{nn=30},CoefficientList[Series[x (1-x)/(1-3x),{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Oct 19 2022 *) -
SageMath
[0,1]+[2*3^(n-2)*factorial(n) for n in (2..30)] # G. C. Greubel, May 31 2022
Formula
E.g.f.: x*(1-x)/(1-3*x)
D-finite recurrence: a(1)=1, a(0)=0, a(2)=4, a(n) = 3*n*a(n-1).
a(n) = 2*3^(n-2)*n! = 2*A153647(n-2), n>1.
From Amiram Eldar, May 31 2025: (Start)
Sum_{n>=1} 1/a(n) = 9*exp(1/3)/2 - 5.
Sum_{n>=1} (-1)^(n+1)/a(n) = 4 - 9*exp(-1/3)/2. (End)