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A052707
Odd powers of 2 multiplied by Catalan numbers.
3
0, 2, 8, 64, 640, 7168, 86016, 1081344, 14057472, 187432960, 2549088256, 35223764992, 493132709888, 6979724509184, 99710350131200, 1435829041889280, 20819521107394560, 303720072625520640, 4454561065174302720
OFFSET
0,2
COMMENTS
And/Or trees with 1 variable [Chauvin et al.]. - R. J. Mathar, Apr 01 2012
LINKS
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.
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)
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 *)
PROG
(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
CROSSREFS
Cf. A000108.
Sequence in context: A353821 A354237 A191570 * A059862 A268666 A193549
KEYWORD
easy,nonn
AUTHOR
encyclopedia(AT)pommard.inria.fr, Jan 25 2000
STATUS
approved