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

0,3

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

Table of n, a(n) for n=0..28.

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

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 *)

Prog

(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

Crossrefs

Cf. A086246, A001006.

Sequence in context: A202135 A068875 A262543 * A289709 A192574 A135336

Adjacent sequences: A348194 A348195 A348196 * A348198 A348199 A348200

Keyword

nonn

Author

Alexander Burstein, Oct 06 2021

Status

approved