A052780 Expansion of e.g.f. x^2*exp(4*x).

0, 0, 2, 24, 192, 1280, 7680, 43008, 229376, 1179648, 5898240, 28835840, 138412032, 654311424, 3053453312, 14092861440, 64424509440, 292057776128, 1314259992576, 5875515260928, 26113401159680

Offset: 0

Author

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

Keywords

• easy
• nonn

Comments

Old name was: A simple grammar.

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 737
• Index entries for linear recurrences with constant coefficients, signature (12,-48,64).

Crossrefs

Cf. A038845.

Programs

GAP

List([0..30], n-> 4^(n-2)*n*(n-1)); # G. C. Greubel, Jul 20 2019

Magma

[4^(n-2)*n*(n-1): n in [0..30]]; // G. C. Greubel, Jul 20 2019

Maple

spec := [S,{B=Set(Z),S=Prod(Z,Z,B,B,B,B)},labeled]: seq(combstruct[count](spec,size=n), n=0..20);
seq(n*(n-1)*4^(n-2), n=0..20); # Zerinvary Lajos, Apr 25 2007

Mathematica

Table[4^(n-2)*n*(n-1), {n,0,30}] (* G. C. Greubel, Jul 20 2019 *)
With[{nn=20},CoefficientList[Series[x^2 Exp[4x],{x,0,nn}],x] Range[0,nn]!] (* or *) LinearRecurrence[{12,-48,64},{0,0,2},30] (* Harvey P. Dale, Sep 28 2022 *)

PARI

vector(30, n, n--; 4^(n-2)*n*(n-1)) \\ G. C. Greubel, Jul 20 2019

Sage

[4^(n-2)*n*(n-1) for n in (0..30)] # G. C. Greubel, Jul 20 2019

Formula

E.g.f.: x^2*exp(x)^4.

a(n) = 2*A038845(n-2).

Recurrence: a(1)=0, a(2)=2, (n-1)*a(n+1) - 4*(n+1)*a(n) = 0.

From Ralf Stephan, Mar 26 2003: (Start)

a(n) = n*(n-1)*4^(n-2).

G.f.: 2*x^2/(1-4*x)^3. (End)

Extensions

New name from e.g.f. by Jon E. Schoenfield, Feb 07 2019