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%I #22 Dec 23 2021 03:01:02
%S 0,1,2,4,10,28,84,264,860,2880,9862,34392,121770,436688,1583146,
%T 5793216,21370806,79391536,296760222,1115327844,4212125662,
%U 15976390684,60833679424,232452408632,891060970152,3425639505624,13204738280326,51024408662932,197607503526934
%N Composition of the g.f. of A086246 with itself.
%C G.f.: A(x) is the pseudo-involutory Riordan companion of 2*M(x)-1, where M(x) is the g.f. of A001006.
%C For 1 <= n <= 7, a(n) coincides with A068875(n-1).
%C Conjecture: a(n) > A068875(n-1) for n > 7 (equivalently, a(n) > 2*A000108(n-1) for n > 7).
%H Alexander Burstein and Louis W. Shapiro, <a href="https://arxiv.org/abs/2112.11595">Pseudo-involutions in the Riordan group</a>, arXiv:2112.11595 [math.CO], 2021.
%F G.f.: A(x) = F(F(x)), where F(x) is the g.f. of A086246.
%F Let G(x) = 2*M(x) - 1, where M(x) is the g.f. of A001006 (equivalently, x*G(x) is the g.f. of A007971). Then G(-A(x)) = 1/G(x).
%F A(-A(x)) = -x.
%F a(n) ~ ((1 + sqrt(3))^(n - 1/2) * 3^(n - 1/2)) / (sqrt(Pi) * n^(3/2) * 2^n). - _Vaclav Kotesovec_, Oct 07 2021
%p gf:= (f-> f(f(x)))(x->(1+x-sqrt(1-2*x-3*x^2))/2):
%p a:= n-> coeff(series(gf,x,n+1),x,n):
%p seq(a(n), n=0..40); # _Alois P. Heinz_, Oct 06 2021
%t f[x_] := (1 + x - Sqrt[1 - 2*x - 3*x^2])/2; a[n_] := SeriesCoefficient[f[f[x]], {x, 0, n}]; Array[a, 25, 0] (* _Amiram Eldar_, Oct 06 2021 *)
%o (PARI) f(x) = (1+x-sqrt(1-2*x-3*x^2))/2;
%o my(x='x+O('x^30)); concat(0, Vec(f(f(x)))) \\ _Michel Marcus_, Oct 06 2021
%Y Cf. A086246, A001006.
%K nonn
%O 0,3
%A _Alexander Burstein_, Oct 06 2021
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