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.

A367919 Expansion of e.g.f. exp(4*(exp(x) - 1) - x).

Original entry on oeis.org

1, 3, 13, 67, 397, 2627, 19085, 150339, 1272205, 11481155, 109852813, 1109011779, 11765211021, 130707706435, 1516160466573, 18314760232771, 229865470694797, 2991427959247939, 40292570823959693, 560791503840522563, 8053114165521427341, 119158887402348541507
Offset: 0

Views

Author

Ilya Gutkovskiy, Dec 04 2023

Keywords

Crossrefs

Cf. A000296, A078944, A217924, A355254, A367891, A367920, A367921.

Programs

  • Mathematica
    nmax = 21; CoefficientList[Series[Exp[4 (Exp[x] - 1) - x], {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = -a[n - 1] + 4 Sum[Binomial[n - 1, k - 1] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 21}]

Formula

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

A367921 Expansion of e.g.f. exp(4*(exp(x) - 1) - 3*x).

Original entry on oeis.org

1, 1, 5, 17, 93, 505, 3269, 22657, 172461, 1407177, 12284629, 113832273, 1114775869, 11487315481, 124118143717, 1401808691489, 16504815145421, 202101235848297, 2568312461002741, 33808677627863537, 460227870278020957, 6468672644291075001, 93745096205219336709
Offset: 0

Views

Author

Ilya Gutkovskiy, Dec 04 2023

Keywords

Crossrefs

Cf. A000296, A078944, A346738, A355253, A367890, A367891, A367919, A367920.

Programs

  • Maple
    b:= proc(n, k, m) option remember; `if`(n=0, 4^m, `if`(k>0,
      b(n-1, k-1, m+1)*k, 0)+m*b(n-1, k, m)+b(n-1, k+1, m))
    end:
a:= n-> b(n, 0$2):
seq(a(n), n=0..22);  # Alois P. Heinz, Apr 29 2025
  • Mathematica
    nmax = 22; 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, 22}]

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.

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