A348189 Pseudo-involutory Riordan companion of 1 + 2*x*M(x), where M(x) is the g.f. of A001006.

1, 0, 0, 2, 0, 6, 8, 24, 60, 148, 396, 1026, 2744, 7350, 19872, 54102, 148104, 407682, 1127328, 3130542, 8726256, 24407634, 68482776, 192698124, 543642476, 1537443024, 4357677516, 12376868254, 35221087656, 100409367690, 286730523104, 820078634232, 2348966799132

Offset: 1

Author

Alexander Burstein, Oct 06 2021

Keywords

• nonn

Links

• Alexander Burstein and Louis W. Shapiro, Pseudo-involutions in the Riordan group, arXiv:2112.11595 [math.CO], 2021.
• Alexander Burstein and Louis W. Shapiro, Pseudo-involutions in the Riordan group and Chebyshev polynomials, arXiv:2502.13673 [math.CO], 2025.

Crossrefs

Cf. A001006, A007971, A086246.

Programs

Mathematica

a[n_] := SeriesCoefficient[(1 - Sqrt[1-2*x-3*x^2])/(x * (2 + x - Sqrt[1-2*x-3*x^2])), {x, 0, n}]; Array[a, 33, 0] (* Amiram Eldar, Oct 06 2021 *)

PARI

my(x='x+O('x^35)); Vec((1-sqrt(1-2*x-3*x^2))/(x*(2+x-sqrt(1-2*x-3*x^2)))) \\ Michel Marcus, Oct 06 2021

Formula

G.f.: A(x) = (1 - sqrt(1 - 2*x - 3*x^2))/(x*(2 + x - sqrt(1 - 2*x - 3*x^2))).

If M(x) is the g.f. of A001006, then A(x) = (1 + 2*x*M(x))/(1 + 2*x + 2*x^2*M(x)).

Let M(x) be the g.f. of A001006 and F(x) = 1 + 2*x*M(x) (equivalently, x*F(x) = g.f. of A007971). Then F(-x*A(x)) = 1/F(x).

A(-x*A(x)) = 1/A(x).

G.f.: Let F(x) be the g.f. of A107264, then A(x) = 1 + 2*x^3*A(x)^2*F(x^2*A(x)). - Alexander Burstein, Feb 14 2022