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.

Showing 1-2 of 2 results.

A328055 Expansion of e.g.f. -log(1 - x / (1 - x)^2).

Original entry on oeis.org

0, 1, 5, 32, 270, 2904, 38400, 605520, 11113200, 232888320, 5488560000, 143704108800, 4138573824000, 130020673305600, 4425201196416000, 162194862064435200, 6369479157000960000, 266808274486161408000, 11874724379464826880000, 559591797303082672128000
Offset: 0

Views

Author

Ilya Gutkovskiy, Oct 03 2019

Keywords

Comments

a(n) is the number of ways to choose one element from each branch of labeled octopuses with n nodes (cf. A029767 and example below). - Enrique Navarrete, Oct 29 2023

Examples

			For n=2, the 3 labeled octopuses are the following, and there are 2+2+1 ways to choose one element from each branch:
O-1-2;
O-2-1;
1-O-2. - _Enrique Navarrete_, Oct 29 2023

Crossrefs

Cf. A001906, A004146, A005248, A005443, A029767, A052567 (exponential transform), A100404, A226968, A328054.

Programs

  • Magma
    [0] cat [Factorial(n - 1)*(Lucas(2*n)-2):n in [1..20]]; // Marius A. Burtea, Oct 03 2019
  • Mathematica
    nmax = 19; CoefficientList[Series[-Log[1 - x/(1 - x)^2], {x, 0, nmax}], x] Range[0, nmax]!
Join[{0}, Table[(n - 1)! (LucasL[2 n] - 2), {n, 1, 19}]]
  • PARI
    my(x='x+O('x^20)); concat(0, Vec(serlaplace(-log(1 - x / (1 - x)^2)))) \\ Michel Marcus, Oct 03 2019

Formula

E.g.f.: log(1 + Sum_{k>=1} Fibonacci(2*k) * x^k).
a(n) = (n - 1)! * (Lucas(2*n) - 2) for n > 0.

A345075 E.g.f.: exp( x*(1 + 2*x) / (1 - x - x^2) ).

Original entry on oeis.org

1, 1, 7, 43, 409, 4441, 58351, 872467, 14776273, 278033329, 5759752951, 130094213371, 3181051122217, 83674165333513, 2355245699211679, 70617410638402531, 2246412316372784161, 75551901666095113057, 2678119105038094325863, 99778611508176786458059, 3897493112463397722989881
Offset: 0

Views

Author

Ilya Gutkovskiy, Jun 07 2021

Keywords

Crossrefs

Cf. A000204, A080833, A100404, A294222.

Programs

  • Maple
    A345075 := proc(n)
    option remember ;
    if n = 0 then
        1;
    else
        add(binomial(n-1,k-1)*k!*procname(n-k)*A000204(k),k=1..n) ;
    end if;
end proc:
seq(A345075(n),n=0..42) ; # R. J. Mathar, Aug 20 2021
  • Mathematica
    nmax = 20; CoefficientList[Series[Exp[x (1 + 2 x)/(1 - x - x^2)], {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = Sum[Binomial[n - 1, k - 1] k! LucasL[k] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 20}]
  • PARI
    my(x='x+O('x^30)); Vec(serlaplace(exp(x*(1+2*x)/(1-x-x^2)))) \\ Michel Marcus, Jun 07 2021

Formula

a(0) = 1; a(n) = Sum_{k=1..n} binomial(n-1,k-1) * k! * Lucas(k) * a(n-k).
a(n) ~ (1 + sqrt(5))^n * exp(1/(2*sqrt(5)) - 1 + 2*sqrt(n) - n) * n^(n - 1/4) / 2^(n + 1/2). - Vaclav Kotesovec, Jun 08 2021
D-finite with recurrence a(n) +(-2*n+1)*a(n-1) -(n+2)*(n-1)*a(n-2) +(2*n-5)*(n-1)*(n-2)*a(n-3) +(n-1)*(n-2)*(n-3)*(n-4)*a(n-4)=0. - R. J. Mathar, Aug 20 2021
Showing 1-2 of 2 results.

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