cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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

Original entry on oeis.org

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

Views

Author

Alexander Burstein, Oct 06 2021

Keywords

Links

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

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

Original entry on oeis.org

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

Views

Author

Alexander Burstein, Oct 06 2021

Keywords

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

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
Previous Showing 11-12 of 12 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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