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.

A355253 Expansion of e.g.f. exp(2*(exp(x) - 1) - 3*x).

Original entry on oeis.org

1, -1, 3, -5, 19, -29, 171, -69, 2339, 5139, 57563, 303403, 2397011, 17237507, 139011211, 1151110299, 10076637827, 91903924979, 874688607035, 8656097294091, 88932728790195, 946748093175523, 10426787247224043, 118620906668843131, 1392128306377939427, 16833088095308098003
Offset: 0

Views

Author

Vaclav Kotesovec, Jun 26 2022

Keywords

Comments

Inverse binomial transform of A194689.

Links

Crossrefs

Cf. A001861, A217923, A194689, A355252.

Programs

  • Mathematica
    nmax = 30; CoefficientList[Series[Exp[2*Exp[x]-2-3*x], {x, 0, nmax}], x] * Range[0, nmax]!
  • PARI
    my(x='x+O('x^30)); Vec(serlaplace(exp(2*(exp(x) - 1) - 3*x))) \\ Michel Marcus, Dec 04 2023

Formula

a(n) ~ 8 * n^(n-3) * exp(n/LambertW(n/2) - n - 2) / (sqrt(1 + LambertW(n/2)) * LambertW(n/2)^(n-3)).
a(0) = 1; a(n) = -3 * a(n-1) + 2 * Sum_{k=1..n} binomial(n-1,k-1) * a(n-k). - Ilya Gutkovskiy, Dec 04 2023

A367937 Expansion of e.g.f. exp(4*(exp(x) - 1) + 3*x).

Original entry on oeis.org

1, 7, 53, 431, 3741, 34471, 335621, 3438943, 36954285, 415187415, 4864054165, 59278367247, 749926582717, 9829744447495, 133267495918885, 1865916660838847, 26942271261464525, 400673643394972983, 6129834703935247285, 96368617886967750767, 1555302323744129219293
Offset: 0

Views

Author

Ilya Gutkovskiy, Dec 05 2023

Keywords

Crossrefs

Cf. A005494, A035009, A078945, A355252, A366199, A367889.

Programs

  • Mathematica
    nmax = 20; CoefficientList[Series[Exp[4 (Exp[x] - 1) + 3 x], {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = 3 a[n - 1] + 4 Sum[Binomial[n - 1, k - 1] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 20}]
  • PARI
    my(x='x+O('x^30)); Vec(serlaplace(exp(4*(exp(x) - 1) + 3*x))) \\ Michel Marcus, Dec 07 2023

Formula

G.f. A(x) satisfies: A(x) = 1 + x * ( 3 * A(x) + 4 * A(x/(1 - x)) / (1 - x) ).
a(n) = exp(-4) * Sum_{k>=0} 4^k * (k+3)^n / k!.
a(0) = 1; a(n) = 3 * a(n-1) + 4 * Sum_{k=1..n} binomial(n-1,k-1) * a(n-k).
Showing 1-2 of 2 results.

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

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