A052760 Expansion of e.g.f.: x^2*(exp(x)-1)^2.

0, 0, 0, 0, 24, 120, 420, 1260, 3472, 9072, 22860, 56100, 134904, 319176, 745108, 1719900, 3931680, 8912352, 20053404, 44825940, 99613960, 220200120, 484441188, 1061157900, 2315254704, 5033163600, 10905189100, 23555209860, 50734299672, 108984793512

Offset: 0

Author

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

Keywords

• easy
• nonn

Comments

Original name: a simple grammar.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 717
• Index entries for linear recurrences with constant coefficients, signature (9,-33,63,-66,36,-8).

Crossrefs

Cf. A052749.

Programs

Magma

[0,0,0] cat [n*(n-1)*(2^n-8)/4: n in [3..30]]; // Vincenzo Librandi, May 05 2013

Maple

spec := [S,{B=Set(Z,1 <= card),S=Prod(B,B,Z,Z)},labeled]: seq(combstruct[count](spec,size=n), n=0..20);

Mathematica

Part[#, Range[1, Length[#], 1]]&@(Array[#!&, Length[#], 0] #)&@CoefficientList[Series[x^2 Exp[x]^2 - 2 Exp[x] x^2 + x^2, {x, 0, 30}], x]//ExpandAll (* Vincenzo Librandi, May 05 2013 *)

PARI

a(n) = if(n<4, 0, n*(n-1)*(2^n-8)/4); \\ Joerg Arndt, May 06 2013

Formula

E.g.f.: x^2*exp(x)^2-2*exp(x)*x^2+x^2.

Recurrence: {a(1)=0, a(2)=0, a(3)=0, a(4)=24, (2*n^2+6*n+4)*a(n)+(6-3*n^2-3*n)*a(n+1)+(n^2-n)*a(n+2)}.

For n>=3, a(n) = n*(n-1)*(2^n-8)/4. - Vaclav Kotesovec, Nov 27 2012

a(n) = n*A052749(n-1) = 2*n*(n-1)*Stirling2(n-2,2) for n >= 2. - Andrew Howroyd, Aug 08 2020

Extensions

More terms from Vincenzo Librandi, May 05 2013

Name changed by Andrew Howroyd, Aug 08 2020