A052555 Expansion of e.g.f. 1/(1-2*x-x^2).

1, 2, 10, 72, 696, 8400, 121680, 2056320, 39715200, 862928640, 20832940800, 553246848000, 16027872537600, 503031194265600, 17001946241280000, 615694938034176000, 23782705115000832000, 976080997055324160000

Offset: 0

Author

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

Keywords

• easy
• nonn

Links

• G. C. Greubel, Table of n, a(n) for n = 0..390
• INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 496

Programs

GAP

a:=[2,10];; for n in [3..20] do a[n]:=2*n*a[n-1]+n*(n-1)*a[n-2]; od; Concatenation([1], a); # G. C. Greubel, May 07 2019

Magma

m:=20; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!( 1/(1-2*x-x^2) )); [Factorial(n-1)*b[n]: n in [1..m]]; // G. C. Greubel, May 07 2019

Maple

spec := [S,{S=Sequence(Union(Z,Z,Prod(Z,Z)))},labeled]: seq(combstruct[count](spec,size=n), n=0..20);
with(combstruct):ZL:=[T,{T=Union(Z,Prod(Epsilon,Z,T),Prod(T,Z,Epsilon),Prod(T,Z,Z))},labeled]:seq(count(ZL,size=i)/i,i=1..18); # Zerinvary Lajos, Dec 16 2007

Mathematica

With[{m = 20}, CoefficientList[Series[1/(1-2*x-x^2), {x,0,m}], x]* Range[0, m]!] (* G. C. Greubel, May 07 2019 *)

PARI

my(x='x+O('x^20)); Vec(serlaplace( 1/(1-2*x-x^2) )) \\ G. C. Greubel, May 07 2019

Sage

m = 20; T = taylor(1/(1-2*x-x^2), x, 0, m); [factorial(n)*T.coefficient(x, n) for n in (0..m)] # G. C. Greubel, May 07 2019

Formula

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

a(n) = 2*n*a(n-1) + n*(n-1)*a(n-2), with a(0)=1, a(1)=2.

a(n) = Sum(1/4*(1+_alpha)*_alpha^(-1-n), _alpha=RootOf(-1+2*_Z+_Z^2))*n!

a(n) = n!*A000129(n+1). - R. J. Mathar, Nov 27 2011