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A052707 Odd powers of 2 multiplied by Catalan numbers. 2
0, 2, 8, 64, 640, 7168, 86016, 1081344, 14057472, 187432960, 2549088256, 35223764992, 493132709888, 6979724509184, 99710350131200, 1435829041889280, 20819521107394560, 303720072625520640, 4454561065174302720 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

And/Or trees with 1 variable [Chauvin et al.]. - R. J. Mathar, Apr 01 2012

LINKS

Table of n, a(n) for n=0..18.

B. Chauvin, P. Flajolet et al., And/Or Tree Revisited, Combinat., Probal. Comput. 13 (2004) 475-497

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 662

FORMULA

a(n-1) = 2^(2*n+1)*A000108(n).

Recurrence: a(1)=2, (-8+16*n)*a(n)-(n+1)*a(n+1) =0.

a(n) = 16^n*(GAMMA(n-1/2)/(8*GAMMA(n+1)*Pi^(1/2))), n>0.

Given g.f. A(x), then B(x)=A(x)-x series reversion is -B(-x). - Michael Somos, Sep 08 2005

Given g.f. A(x), then B(x)=A(x)-x satisfies B(x)=x+8*C(16*x*B(x)) where C(x) is g.f. for Catalan number A000108.

G.f. A(x) = 2*x*C(4*x) where C(x) is g.f. for Catalan number A000108.

G.f.: (1-sqrt(1-16*x))/4 = 4*x/(1+sqrt(1-16*x)).

MAPLE

spec := [S, {C=Union(B, Z), B=Prod(S, S), S=Union(B, C, Z)}, unlabeled]: seq(combstruct[count](spec, size=n), n=0..20);

# 2nd program

A052707 := proc(n)

    if n =0 then

        0;

    else

        2^(2*n-1)*A000108(n-1) ;

    fi ;

end proc:

seq(A052707(n), n=0..10) ; # R. J. Mathar, Apr 26 2017

MATHEMATICA

InverseSeries[Series[y/2-y^2, {y, 0, 24}], x] (* then A(x)=y(x) *) (* Len Smiley, Apr 13 2000 *)

PROG

(PARI) a(n)=if(n<1, 0, n--; 2*4^n*binomial(2*n, n)/(n+1))

CROSSREFS

Sequence in context: A287229 A323853 A191570 * A059862 A268666 A193549

Adjacent sequences:  A052704 A052705 A052706 * A052708 A052709 A052710

KEYWORD

easy,nonn

AUTHOR

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

STATUS

approved

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Last modified September 16 08:36 EDT 2019. Contains 327091 sequences. (Running on oeis4.)