A052707 Odd powers of 2 multiplied by Catalan numbers.

0, 2, 8, 64, 640, 7168, 86016, 1081344, 14057472, 187432960, 2549088256, 35223764992, 493132709888, 6979724509184, 99710350131200, 1435829041889280, 20819521107394560, 303720072625520640, 4454561065174302720

Offset: 0

Author

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

Keywords

• easy
• nonn

Comments

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

Links

• G. C. Greubel, Table of n, a(n) for n = 0..820
• B. Chauvin, P. Flajolet, D. Gardy, and B. Gittenberger, And/Or Tree Revisited, Combinatorics, Probability and Computing, Vol. 13, No. 4-5 (2004), pp. 475-497.
• INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 662.

Crossrefs

Cf. A000108.

Programs

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 *)
Table[2^(2*n-1)*CatalanNumber[n-1] +Boole[n==0]/2, {n, 0, 30}] (* G. C. Greubel, May 30 2022 *)

PARI

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

SageMath

[2^(2*n-1)*catalan_number(n-1) + bool(n==0)/4 for n in (0..30)] # G. C. Greubel, May 30 2022

Formula

a(n) = 2^(2*n-1)*A000108(n-1), for n > 0.

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

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)).

From Amiram Eldar, Mar 22 2022: (Start)

Sum_{n>=1} 1/a(n) = (4/75)*(11 + 16*arcsin(1/4)/sqrt(15)).

Sum_{n>=1} (-1)^(n+1)/a(n) = (4/4913)*(527 - 48*sqrt(17)*arcsinh(1/4)). (End)