A348197 Composition of the g.f. of A086246 with itself.

0, 1, 2, 4, 10, 28, 84, 264, 860, 2880, 9862, 34392, 121770, 436688, 1583146, 5793216, 21370806, 79391536, 296760222, 1115327844, 4212125662, 15976390684, 60833679424, 232452408632, 891060970152, 3425639505624, 13204738280326, 51024408662932, 197607503526934

Offset: 0

Author

Alexander Burstein, Oct 06 2021

Keywords

• nonn

Comments

G.f.: A(x) is the pseudo-involutory Riordan companion of 2*M(x)-1, where M(x) is the g.f. of A001006.

For 1 <= n <= 7, a(n) coincides with A068875(n-1).

Conjecture: a(n) > A068875(n-1) for n > 7 (equivalently, a(n) > 2*A000108(n-1) for n > 7).

Links

• Alexander Burstein and Louis W. Shapiro, Pseudo-involutions in the Riordan group, arXiv:2112.11595 [math.CO], 2021.

Crossrefs

Cf. A086246, A001006.

Programs

Maple

gf:= (f-> f(f(x)))(x->(1+x-sqrt(1-2*x-3*x^2))/2):
a:= n-> coeff(series(gf,x,n+1),x,n):
seq(a(n), n=0..40);  # Alois P. Heinz, Oct 06 2021

Mathematica

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 *)

PARI

f(x) = (1+x-sqrt(1-2*x-3*x^2))/2;
my(x='x+O('x^30)); concat(0, Vec(f(f(x)))) \\ Michel Marcus, Oct 06 2021

Formula

G.f.: A(x) = F(F(x)), where F(x) is the g.f. of A086246.

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).

A(-A(x)) = -x.

a(n) ~ ((1 + sqrt(3))^(n - 1/2) * 3^(n - 1/2)) / (sqrt(Pi) * n^(3/2) * 2^n). - Vaclav Kotesovec, Oct 07 2021