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

0,3

Links

G. C. Greubel, Table of n, a(n) for n = 0..375

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 650

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)

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 *)

Prog

(SageMath) [0, 1]+[2*3^(n-2)*factorial(n) for n in (2..30)] # G. C. Greubel, May 31 2022

Crossrefs

Cf. A153647.

Sequence in context: A371679 A132864 A294050 * A167540 A387714 A374857

Adjacent sequences: A052697 A052698 A052699 * A052701 A052702 A052703

Keyword

easy,nonn

Author

INRIA Encyclopedia of Combinatorial Structures, Jan 25 2000

Status

approved