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A052727 Expansion of e.g.f. 1/2-1/2*(1-4*x-4*x^2)^(1/2).

%I A052727 #17 Jan 13 2025 08:50:44
%S A052727 0,1,4,24,288,4800,103680,2741760,85800960,3100446720,127037030400,
%T A052727 5819550105600,294727768473600,16350861400473600,986127353590579200,
%U A052727 64238655955009536000,4495021381191204864000,336249161369543245824000
%N A052727 Expansion of e.g.f. 1/2-1/2*(1-4*x-4*x^2)^(1/2).
%C A052727 Old name was: A simple context-free grammar in a labeled universe.
%H A052727 INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=683">Encyclopedia of Combinatorial Structures 683</a>
%F A052727 Recurrence: {a(1)=1, a(2)=4, (-4*n^2+4)*a(n) +(-4*n-2)*a(n+1) +a(n+2) =0.
%F A052727 a(n) ~ sqrt(2-sqrt(2))* ((1+sqrt(2))/exp(1))^n * (2*n)^(n-1). - _Vaclav Kotesovec_, Sep 30 2013
%F A052727 a(n) = n!*A025227(n). - _R. J. Mathar_, Oct 18 2013
%p A052727 spec := [S,{B=Prod(S,S),S=Union(B,Z,C),C=Prod(Z,Z)},labeled]: seq(combstruct[count](spec,size=n), n=0..20);
%t A052727 CoefficientList[Series[1/2-1/2*(1-4*x-4*x^2)^(1/2), {x, 0, 20}], x]* Range[0, 20]! (* _Vaclav Kotesovec_, Sep 30 2013 *)
%K A052727 easy,nonn
%O A052727 0,3
%A A052727 encyclopedia(AT)pommard.inria.fr, Jan 25 2000