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

1,4

Links

Table of n, a(n) for n=1..33.

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

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 *)

Prog

(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

Crossrefs

Cf. A001006, A007971, A086246.

Sequence in context: A334349 A058498 A358167 * A372911 A003076 A175478

Adjacent sequences: A348186 A348187 A348188 * A348190 A348191 A348192

Keyword

nonn

Author

Alexander Burstein, Oct 06 2021

Status

approved