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

%I A052780 #29 Sep 28 2022 16:44:23
%S A052780 0,0,2,24,192,1280,7680,43008,229376,1179648,5898240,28835840,
%T A052780 138412032,654311424,3053453312,14092861440,64424509440,292057776128,
%U A052780 1314259992576,5875515260928,26113401159680
%N A052780 Expansion of e.g.f. x^2*exp(4*x).
%C A052780 Old name was: A simple grammar.
%H A052780 G. C. Greubel, <a href="/A052780/b052780.txt">Table of n, a(n) for n = 0..1000</a>
%H A052780 INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=737">Encyclopedia of Combinatorial Structures 737</a>
%H A052780 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (12,-48,64).
%F A052780 E.g.f.: x^2*exp(x)^4.
%F A052780 a(n) = 2*A038845(n-2).
%F A052780 Recurrence: a(1)=0, a(2)=2, (n-1)*a(n+1) - 4*(n+1)*a(n) = 0.
%F A052780 From _Ralf Stephan_, Mar 26 2003: (Start)
%F A052780 a(n) = n*(n-1)*4^(n-2).
%F A052780 G.f.: 2*x^2/(1-4*x)^3. (End)
%p A052780 spec := [S,{B=Set(Z),S=Prod(Z,Z,B,B,B,B)},labeled]: seq(combstruct[count](spec,size=n), n=0..20);
%p A052780 seq(n*(n-1)*4^(n-2), n=0..20); # _Zerinvary Lajos_, Apr 25 2007
%t A052780 Table[4^(n-2)*n*(n-1), {n,0,30}] (* _G. C. Greubel_, Jul 20 2019 *)
%t A052780 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 *)
%o A052780 (PARI) vector(30, n, n--; 4^(n-2)*n*(n-1)) \\ _G. C. Greubel_, Jul 20 2019
%o A052780 (Magma) [4^(n-2)*n*(n-1): n in [0..30]]; // _G. C. Greubel_, Jul 20 2019
%o A052780 (Sage) [4^(n-2)*n*(n-1) for n in (0..30)] # _G. C. Greubel_, Jul 20 2019
%o A052780 (GAP) List([0..30], n-> 4^(n-2)*n*(n-1)); # _G. C. Greubel_, Jul 20 2019
%Y A052780 Cf. A038845.
%K A052780 easy,nonn
%O A052780 0,3
%A A052780 encyclopedia(AT)pommard.inria.fr, Jan 25 2000
%E A052780 New name from e.g.f. by _Jon E. Schoenfield_, Feb 07 2019