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 1-3 of 3 results.

A321989 Expansion of e.g.f. exp(exp(-x)/(1 - x) - 1).

Original entry on oeis.org

1, 0, 1, 2, 12, 64, 455, 3618, 33131, 338728, 3838572, 47678520, 644172045, 9402091620, 147405489205, 2470129035710, 44053120590540, 833000495161600, 16644648834503555, 350406040769989974, 7751328201878523295, 179738821179613739780, 4359334293132050359932, 110368937036048741434824
Offset: 0

Views

Author

Ilya Gutkovskiy, Dec 19 2018

Keywords

Crossrefs

Cf. A000166, A000262, A318365, A321974.

Programs

  • Maple
    seq(coeff(series(factorial(n)*exp(exp(-x)/(1-x)-1),x,n+1), x, n), n = 0 .. 25); # Muniru A Asiru, Dec 19 2018
  • Mathematica
    nmax = 23; CoefficientList[Series[Exp[Exp[-x]/(1 - x) - 1], {x, 0, nmax}], x] Range[0, nmax]!
a[n_] := a[n] = Sum[Subfactorial[k] Binomial[n - 1, k - 1] a[n - k], {k, n}]; a[0] = 1; Table[a[n], {n, 0, 23}]

Formula

a(0) = 1; a(n) = Sum_{k=1..n} A000166(k)*binomial(n-1,k-1)*a(n-k).
a(n) ~ exp(3*exp(-1)/2 - 5/4 + 2*exp(-1/2)*sqrt(n) - n) * n^(n-1/4) / sqrt(2). - Vaclav Kotesovec, Dec 19 2018

A308330 a(n) = n! * [x^n] exp(exp(n*x)/(1 - x) - 1).

Original entry on oeis.org

1, 2, 19, 346, 10217, 441226, 26023123, 1998840586, 193094418161, 22841006706928, 3239088790361491, 541309430523114804, 105106521730010262745, 23431755937256853296514, 5936989025261397848036755, 1694791457312643753292004446, 540937403928198054978670965089
Offset: 0

Views

Author

Ilya Gutkovskiy, May 20 2019

Keywords

Crossrefs

Cf. A063170, A292914, A308331, A321974.

Programs

  • Mathematica
    Table[n! SeriesCoefficient[Exp[Exp[n x]/(1 - x) - 1], {x, 0, n}], {n, 0, 16}]

A328008 Expansion of e.g.f. 1 / (2 - exp(x) / (1 - x)).

Original entry on oeis.org

1, 2, 13, 124, 1575, 25006, 476421, 10589720, 269010979, 7687905826, 244120131393, 8526912775756, 324914136199263, 13412430958497494, 596253684006657085, 28399969571266895488, 1442890578572155475355, 77889310498718258171914, 4451905168738601015593785
Offset: 0

Views

Author

Ilya Gutkovskiy, Oct 01 2019

Keywords

Crossrefs

Cf. A000522, A002866, A321974, A328007.

Programs

  • Mathematica
    nmax = 18; CoefficientList[Series[1/(2 - Exp[x]/(1 - x)), {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = Sum[Binomial[n, k] Floor[Exp[1] k!] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 18}]
  • PARI
    my(x='x+O('x^25)); Vec(serlaplace(1/(2-exp(x)/(1-x)))) \\ Michel Marcus, Oct 02 2019

Formula

a(0) = 1; a(n) = Sum_{k=1..n} binomial(n,k) * A000522(k) * a(n-k).
a(n) ~ n! / (2*(1 + 1/LambertW(exp(1)/2)) * (1 - LambertW(exp(1)/2))^(n+1)). - Vaclav Kotesovec, Oct 02 2019
Showing 1-3 of 3 results.

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