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.

A331618 E.g.f.: exp(1 / (1 - arctanh(x)) - 1).

Original entry on oeis.org

1, 1, 3, 15, 97, 785, 7523, 83615, 1053281, 14838177, 230832867, 3929944623, 72633052545, 1447981700529, 30960823851267, 706676217730239, 17145815895371073, 440594781536265537, 11952178787661839427, 341291300477569866831, 10231558345117929439521
Offset: 0

Views

Author

Ilya Gutkovskiy, Jan 22 2020

Keywords

Crossrefs

Cf. A000246, A080833, A202139, A296676, A331610, A331615, A331616, A331617.

Programs

  • Mathematica
    nmax = 20; CoefficientList[Series[Exp[1/(1 - ArcTanh[x]) - 1], {x, 0, nmax}], x] Range[0, nmax]!
A296676[0] = 1; A296676[n_] := A296676[n] = Sum[Binomial[n, k] If[OddQ[k], (k - 1)!, 0] A296676[n - k], {k, 1, n}]; a[0] = 1; a[n_] := a[n] = Sum[Binomial[n - 1, k - 1] A296676[k] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 20}]
  • PARI
    seq(n)={Vec(serlaplace(exp(1/(1 - atanh(x + O(x*x^n))) - 1)))} \\ Andrew Howroyd, Jan 22 2020

Formula

a(0) = 1; a(n) = Sum_{k=1..n} binomial(n-1,k-1) * A296676(k) * a(n-k).
a(n) ~ (exp(2) + 1)^(n - 1/4) * n^(n - 1/4) / ((exp(2) - 1)^(n + 1/4) * exp(n - 4*exp(1)*sqrt(n/(exp(4) - 1)) - 2/(exp(4) - 1) - 1/2)). - Vaclav Kotesovec, Jan 26 2020

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.

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

Content is available under The OEIS End-User License Agreement.