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.

A321974 Expansion of e.g.f. exp(exp(x)/(1 - x) - 1).

Original entry on oeis.org

1, 2, 9, 54, 404, 3598, 37003, 430300, 5571147, 79358032, 1231990840, 20684884234, 373208232229, 7197079035318, 147658793214733, 3210107125516682, 73690798853163884, 1780718798351625094, 45171972342078432287, 1199948465249850848608, 33305064129201851432591, 963911863209583899492324
Offset: 0

Views

Author

Ilya Gutkovskiy, Dec 19 2018

Keywords

Crossrefs

Cf. A000262, A000522, A318364, A321989.

Programs

  • Maple
    seq(n!*coeff(series(exp(exp(x)/(1 - x) - 1), x=0, 22), x, n), n=0..21); # Paolo P. Lava, Jan 09 2019
  • Mathematica
    nmax = 21; CoefficientList[Series[Exp[Exp[x]/(1 - x) - 1], {x, 0, nmax}], x] Range[0, nmax]!
a[n_] := a[n] = Sum[Floor[Exp[1] k!] Binomial[n - 1, k - 1] a[n - k], {k, n}]; a[0] = 1; Table[a[n], {n, 0, 21}]

Formula

a(0) = 1; a(n) = Sum_{k=1..n} A000522(k)*binomial(n-1,k-1)*a(n-k).
a(n) ~ exp(-exp(1)/2 - 3/4 + 2*exp(1/2)*sqrt(n) - n) * n^(n - 1/4) / sqrt(2). - Vaclav Kotesovec, Dec 19 2018

A308331 a(n) = n! * [x^n] exp(exp(n*x)/(1 + x) - 1).

Original entry on oeis.org

1, 0, 3, 50, 1449, 61724, 3608515, 275520972, 26505128433, 3125830471928, 442286373458691, 73789189395157730, 14309059313820886681, 3186711239965235356776, 806772967716453793227523, 230153293624841114893344854, 73420355768107554901016231265
Offset: 0

Views

Author

Ilya Gutkovskiy, May 20 2019

Keywords

Crossrefs

Cf. A134095, A292914, A308330, A321989.

Programs

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

A328007 Expansion of e.g.f. 1 / (2 - exp(-x) / (1 - x)).

Original entry on oeis.org

1, 0, 1, 2, 15, 84, 705, 6222, 65779, 765608, 9999333, 143009250, 2235857943, 37833382716, 689729792713, 13469761663862, 280613761282875, 6211105772020560, 145566258957724845, 3601055676894146442, 93772841089130278495, 2563969299245947753700, 73443322391840827563921
Offset: 0

Views

Author

Ilya Gutkovskiy, Oct 01 2019

Keywords

Crossrefs

Cf. A000166, A002866, A302557, A321989, A328008.

Programs

  • Mathematica
    nmax = 22; 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] Subfactorial[k] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 22}]
  • 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) * A000166(k) * a(n-k).
a(n) ~ n! * (-LambertW(-exp(-1)/2) / (2*(1 + LambertW(-exp(-1)/2))^(n+2))). - Vaclav Kotesovec, Oct 02 2019
Showing 1-3 of 3 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.