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A052707 Odd powers of 2 multiplied by Catalan numbers.

%I A052707 #47 Sep 22 2023 07:45:28
%S A052707 0,2,8,64,640,7168,86016,1081344,14057472,187432960,2549088256,
%T A052707 35223764992,493132709888,6979724509184,99710350131200,
%U A052707 1435829041889280,20819521107394560,303720072625520640,4454561065174302720
%N A052707 Odd powers of 2 multiplied by Catalan numbers.
%C A052707 And/Or trees with 1 variable [Chauvin et al.]. - _R. J. Mathar_, Apr 01 2012
%H A052707 G. C. Greubel, <a href="/A052707/b052707.txt">Table of n, a(n) for n = 0..820</a>
%H A052707 B. Chauvin, P. Flajolet, D. Gardy, and B. Gittenberger, <a href="http://dx.doi.org/10.1017/S0963548304006273">And/Or Tree Revisited</a>, Combinatorics, Probability and Computing, Vol. 13, No. 4-5 (2004), pp. 475-497.
%H A052707 INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=662">Encyclopedia of Combinatorial Structures 662</a>.
%F A052707 a(n) = 2^(2*n-1)*A000108(n-1), for n > 0.
%F A052707 Recurrence: a(1)=2, (n+1)*a(n+1) = 8*(2*n-1)*a(n).
%F A052707 a(n) = 16^n*(Gamma(n-1/2)/(8*Gamma(n+1)*Pi^(1/2))), n>0.
%F A052707 Given g.f. A(x), then B(x)=A(x)-x series reversion is -B(-x). - _Michael Somos_, Sep 08 2005
%F A052707 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.
%F A052707 G.f. A(x) = 2*x*C(4*x) where C(x) is g.f. for Catalan number A000108.
%F A052707 G.f.: (1-sqrt(1-16*x))/4 = 4*x/(1+sqrt(1-16*x)).
%F A052707 From _Amiram Eldar_, Mar 22 2022: (Start)
%F A052707 Sum_{n>=1} 1/a(n) = (4/75)*(11 + 16*arcsin(1/4)/sqrt(15)).
%F A052707 Sum_{n>=1} (-1)^(n+1)/a(n) = (4/4913)*(527 - 48*sqrt(17)*arcsinh(1/4)). (End)
%p A052707 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);
%p A052707 # 2nd program
%p A052707 A052707 := proc(n)
%p A052707     if n =0 then
%p A052707         0;
%p A052707     else
%p A052707         2^(2*n-1)*A000108(n-1) ;
%p A052707     fi ;
%p A052707 end proc:
%p A052707 seq(A052707(n),n=0..10) ; # _R. J. Mathar_, Apr 26 2017
%t A052707 InverseSeries[Series[y/2-y^2, {y, 0, 24}], x] (* then A(x)=y(x) *) (* _Len Smiley_, Apr 13 2000 *)
%t A052707 Table[2^(2*n-1)*CatalanNumber[n-1] +Boole[n==0]/2, {n, 0, 30}] (* _G. C. Greubel_, May 30 2022 *)
%o A052707 (PARI) a(n)=if(n<1, 0, n--; 2*4^n*binomial(2*n,n)/(n+1))
%o A052707 (SageMath) [2^(2*n-1)*catalan_number(n-1) + bool(n==0)/4 for n in (0..30)] # _G. C. Greubel_, May 30 2022
%Y A052707 Cf. A000108.
%K A052707 easy,nonn
%O A052707 0,2
%A A052707 encyclopedia(AT)pommard.inria.fr, Jan 25 2000