A052644 Expansion of e.g.f. (1+3x-3x^2)/(1-x)^2.

1, 5, 12, 42, 192, 1080, 7200, 55440, 483840, 4717440, 50803200, 598752000, 7664025600, 105859353600, 1569209241600, 24845812992000, 418455797760000, 7469435990016000, 140852221526016000

Offset: 0

Author

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

Keywords

• easy
• nonn

Links

• INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 590

Crossrefs

Cf. sequences with formula (n + k)*n! listed in A282466.

Programs

Maple

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

Mathematica

With[{nn=20},CoefficientList[Series[(1+3x-3x^2)/(1-x)^2,{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Nov 06 2014 *)

Formula

E.g.f.: -(-3*x+3*x^2-1)/(-1+x)^2

Recurrence: {a(0)=1, a(1)=5, (-n^2-6*n-5)*a(n)+(n+4)*a(n+1)=0, a(2)=12}.

a(n) = (n+4)*n!, n>0.

G.f.: G(0) where G(k) = 1 + x*(k+1)*(k+4)/(1 - 1/(1 + (k+4)/G(k+1))); (continued fraction, 3-step). - Sergei N. Gladkovskii, Oct 16 2012

From Amiram Eldar, Nov 06 2020: (Start)

Sum_{n>=0} 1/a(n) = 27/4- 2*e.

Sum_{n>=0} (-1)^n/a(n) = 27/4 - 16/e. (End)