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.

A367891 Expansion of e.g.f. exp(4*(exp(x) - 1 - x)).

Original entry on oeis.org

1, 0, 4, 4, 52, 164, 1364, 7620, 60148, 449252, 3831700, 33811716, 320082228, 3178774564, 33234163668, 363535920196, 4153091085172, 49406896240996, 610777358429204, 7830140410294148, 103914148870277556, 1425254885630973604, 20173671034640405588
Offset: 0

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Author

Ilya Gutkovskiy, Dec 04 2023

Keywords

Crossrefs

Cf. A000296, A078944, A194689, A346739, A367890.

Programs

  • Mathematica
    nmax = 22; CoefficientList[Series[Exp[4 (Exp[x] - 1 - x)], {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = 4 Sum[Binomial[n - 1, k] a[n - k - 1], {k, 1, n - 1}]; Table[a[n], {n, 0, 22}]
Table[Sum[Binomial[n, k] (-4)^(n - k) BellB[k, 4], {k, 0, n}], {n, 0, 22}]
  • PARI
    my(x='x+O('x^30)); Vec(serlaplace(exp(4*(exp(x) - 1 - x)))) \\ Michel Marcus, Dec 04 2023

Formula

G.f. A(x) satisfies: A(x) = 1 - 4 * x * ( A(x) - A(x/(1 - x)) / (1 - x) ).
a(n) = exp(-4) * Sum_{k>=0} 4^k * (k-4)^n / k!.
a(0) = 1; a(n) = 4 * Sum_{k=1..n-1} binomial(n-1,k) * a(n-k-1).
a(n) = Sum_{k=0..n} binomial(n,k) * (-4)^(n-k) * A078944(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.

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