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.

A320258 a(n) = n! * [x^n] exp(x*exp(-n*x)).

Original entry on oeis.org

1, 1, -3, 10, 81, -4724, 156205, -4406814, 76958273, 3775676248, -698309272899, 72802616429830, -6310377003297455, 435451735391849892, -10028808876450831571, -4757293711381352201774, 1464955115044140633346305, -310063138309576689774123728, 55179706013436631385620675837
Offset: 0

Views

Author

Ilya Gutkovskiy, Oct 08 2018

Keywords

Crossrefs

Cf. A003725, A295552.

Programs

  • Mathematica
    Table[n! SeriesCoefficient[Exp[x Exp[-n x]], {x, 0, n}], {n, 0, 18}]
Table[SeriesCoefficient[Sum[x^k/(1 + n k x)^(k + 1), {k, 0, n}], {x, 0, n}], {n, 0, 18}]
Join[{1}, Table[Sum[Binomial[n, k] (-n k)^(n - k), {k, 0, n}], {n, 18}]]

Formula

a(n) = [x^n] Sum_{k>=0} x^k/(1 + n*k*x)^(k+1).
a(n) = Sum_{k=0..n} binomial(n,k)*(-n*k)^(n-k).

A356827 Expansion of e.g.f. exp(x * exp(3*x)).

Original entry on oeis.org

1, 1, 7, 46, 361, 3436, 37729, 463366, 6280369, 93015352, 1491337441, 25684077706, 472217487625, 9221588527204, 190441412508481, 4143470377262806, 94663498086222049, 2264440394856702832, 56570146384760433217, 1472545685988162638722
Offset: 0

Views

Author

Seiichi Manyama, Aug 29 2022

Keywords

Crossrefs

Cf. A000248, A003725, A216689, A295552.
Cf. A277456, A336951, A351737, A355501, A356820.

Programs

  • Maple
    A356827 := proc(n)
    add((3*k)^(n-k) * binomial(n,k),k=0..n) ;
end proc:
seq(A356827(n),n=0..70) ; # R. J. Mathar, Dec 04 2023
  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(x*exp(3*x))))
  • PARI
    my(N=20, x='x+O('x^N)); Vec(sum(k=0, N, x^k/(1-3*k*x)^(k+1)))
  • PARI
    a(n) = sum(k=0, n, (3*k)^(n-k)*binomial(n, k));

Formula

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

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