A052563 Expansion of e.g.f.: (1-x)/(1-3*x).

1, 2, 12, 108, 1296, 19440, 349920, 7348320, 176359680, 4761711360, 142851340800, 4714094246400, 169707392870400, 6618588321945600, 277980709521715200, 12509131928477184000, 600438332566904832000, 30622354960912146432000, 1653607167889255907328000

Offset

0,2

Comments

Laguerre transform of A052585. - Paul Barry, Aug 08 2008

Links

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

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 505.

Formula

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

Recurrence: a(0) = 1, a(1) = 2, (-3*n-3)*a(n)+a(n+1) = 0.

a(n) = 2*3^(n-1)*n!, for n >= 1.

a(n) = Sum_{k=0..n} binomial(n,k)*(n!/k!)*k!*A001045(k+1). - Paul Barry, Aug 08 2008

From Amiram Eldar, Dec 23 2025: (Start)

Sum_{n>=0} 1/a(n) = (3*exp(1/3)-1)/2.

Sum_{n>=0} (-1)^n/a(n) = (3*exp(-1/3)-1)/2. (End)

Maple

spec := [S, {S=Sequence(Prod(Union(Z, Z), Sequence(Z)))}, labeled]: seq(combstruct[count](spec, size=n), n=0..20);

Mathematica

With[{nn=20}, CoefficientList[Series[(1-x)/(1-3x), {x, 0, nn}], x] Range[ 0, nn]!] (* Harvey P. Dale, May 21 2014 *)
a[n_] := 2*3^(n-1)*n!; a[0] = 1; Array[a, 20, 0] (* Amiram Eldar, Dec 23 2025 *)

Prog

(PARI) my(x='x+O('x^30)); Vec(serlaplace((1-x)/(1-3*x))) \\ G. C. Greubel, May 23 2018
(Magma) m:=25; R<x>:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!((1-x)/(1-3*x))); [Factorial(n-1)*b[n]: n in [1..m]]; // G. C. Greubel, May 23 2018

Crossrefs

Cf. A001045, A052585.

Sequence in context: A212273 A055897 A210997 * A316704 A228173 A218652

Adjacent sequences: A052560 A052561 A052562 * A052564 A052565 A052566

Keyword

easy,nonn

Author

INRIA Encyclopedia of Combinatorial Structures, Jan 25 2000

Status

approved