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

0, 1, 4, 24, 288, 4800, 103680, 2741760, 85800960, 3100446720, 127037030400, 5819550105600, 294727768473600, 16350861400473600, 986127353590579200, 64238655955009536000, 4495021381191204864000, 336249161369543245824000

Offset: 0

Author

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

Keywords

• easy
• nonn

Comments

Old name was: A simple context-free grammar in a labeled universe.

Links

• INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 683

Programs

Maple

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);

Mathematica

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 *)

Formula

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

a(n) ~ sqrt(2-sqrt(2))* ((1+sqrt(2))/exp(1))^n * (2*n)^(n-1). - Vaclav Kotesovec, Sep 30 2013

a(n) = n!*A025227(n). - R. J. Mathar, Oct 18 2013