A052994 Expansion of 2x(1-x)/(1-2x-x^2+x^3).

0, 2, 2, 6, 12, 28, 62, 140, 314, 706, 1586, 3564, 8008, 17994, 40432, 90850, 204138, 458694, 1030676, 2315908, 5203798, 11692828, 26273546, 59036122, 132652962, 298068500, 669753840, 1504923218, 3381531776, 7598232930, 17073074418

Offset: 0

Author

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

Keywords

• easy
• nonn

Links

• INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 1071
• Sergey Kitaev, Jeffrey Remmel, (a,b)-rectangle patterns in permutations and words, arXiv:1304.4286 [math.CO], 2013.
• Index entries for linear recurrences with constant coefficients, signature (2,1,-1).

Crossrefs

Equals 2 * A006356(n-2), n>1.

Programs

Maple

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

PARI

concat(0, Vec(-2*x*(-1+x)/(x^3-x^2-2*x+1) + O(x^40))) \\ Michel Marcus, Mar 19 2015

Formula

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

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

Sum(2/7*(-_alpha+_alpha^2+1)*_alpha^(-1-n), _alpha=RootOf(_Z^3-_Z^2-2*_Z+1))

Extensions

More terms from James Sellers, Jun 05 2000