A348189 - OEIS

%I #18 Feb 15 2022 11:12:00

%S 1,0,0,2,0,6,8,24,60,148,396,1026,2744,7350,19872,54102,148104,407682,

%T 1127328,3130542,8726256,24407634,68482776,192698124,543642476,

%U 1537443024,4357677516,12376868254,35221087656,100409367690,286730523104,820078634232,2348966799132

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

%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) = (1 - sqrt(1 - 2*x - 3*x^2))/(x*(2 + x - sqrt(1 - 2*x - 3*x^2))).

%F 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)).

%F 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).

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

%F 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

%t 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 *)

%o (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

%Y Cf. A001006, A007971, A086246.

%K nonn

%O 1,4

%A _Alexander Burstein_, Oct 06 2021