A052611 Expansion of e.g.f. 1/(1-2*x-2*x^2).

1, 2, 12, 96, 1056, 14400, 236160, 4515840, 98703360, 2426941440, 66305433600, 1992646656000, 65328154214400, 2320237766246400, 88746105588940800, 3636883029491712000, 158978387626426368000, 7383729547341987840000

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..380
• INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 556

Crossrefs

Cf. A002605, A080599.

Programs

Magma

[n le 2 select n else 2*(n-1)*Self(n-1) + 2*(n-1)*(n-2)*Self(n-2): n in [1..41]]; // G. C. Greubel, Jan 31 2023

Maple

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

Mathematica

With[{nn=20},CoefficientList[Series[1/(1-2x-2x^2),{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Apr 14 2015 *)
Table[n!*(-I*Sqrt[2])^(n)*ChebyshevU[n,I/Sqrt[2]], {n,0,40}] (* G. C. Greubel, Jan 31 2023 *)

SageMath

A002605=BinaryRecurrenceSequence(2,2,0,1)
def A052611(n): return factorial(n)*A002605(n+1)
[A052611(n) for n in range(41)] # G. C. Greubel, Jan 31 2023

Formula

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

a(n) = 2^n * A080599(n).

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

a(n) = (n!/6) * Sum_{p = RootOf(2*z^2+2*z-1)} (1+2*p)*p^(-n-1).

a(n) = n!*A002605(n+1). - R. J. Mathar, Nov 27 2011