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.

Previous Showing 21-22 of 22 results.

A366199 Expansion of e.g.f. exp(4*(exp(x) - 1) + 2*x).

Original entry on oeis.org

1, 6, 40, 292, 2308, 19580, 177044, 1696572, 17148916, 182114972, 2024979604, 23506175868, 284125820724, 3567957972316, 46454893734612, 625979771144764, 8715626185644916, 125200337417147932, 1853095248414187796, 28225529312569364732, 441925530173009732532
Offset: 0

Views

Author

Ilya Gutkovskiy, Dec 05 2023

Keywords

Crossrefs

Cf. A005493, A035009, A078945, A355247, A367889, A367937.

Programs

  • Mathematica
    nmax = 20; CoefficientList[Series[Exp[4 (Exp[x] - 1) + 2 x], {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = 2 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) + 2*x))) \\ Michel Marcus, Dec 07 2023

Formula

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

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).
Previous Showing 21-22 of 22 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.