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.

A277473 E.g.f.: -exp(x)*LambertW(-x).

Original entry on oeis.org

0, 1, 4, 18, 116, 1060, 12702, 187810, 3296120, 66897288, 1540762010, 39693752494, 1130866726596, 35300006582620, 1198036854980630, 43921652697277170, 1729775120233353968, 72831210167041246480, 3264674481128340280242, 155220967397580333229270
Offset: 0

Views

Author

Vaclav Kotesovec, Oct 17 2016

Keywords

Crossrefs

Partial sums of A038051.

Programs

  • Mathematica
    CoefficientList[Series[-Exp[x]*LambertW[-x], {x, 0, 20}], x] * Range[0, 20]!
    Table[Sum[Binomial[n, k]*k^(k-1), {k, 1, n}], {n, 0, 20}]
  • PARI
    x='x+O('x^50); concat([0], Vec(serlaplace(-exp(x)*lambertw(-x)))) \\ G. C. Greubel, Jun 11 2017

Formula

a(n) = Sum_{k=1..n} binomial(n,k) * k^(k-1).
a(n) ~ exp(exp(-1)) * n^(n-1).

A277485 E.g.f.: -exp(2*x)*LambertW(-x).

Original entry on oeis.org

0, 1, 6, 33, 216, 1865, 21228, 303765, 5222864, 104540337, 2383558740, 60933722069, 1725392415288, 53590463856345, 1811281159509500, 66172416761172885, 2598298697830360992, 109116931783034360801, 4880122696811960470692, 231565260558289051906965
Offset: 0

Views

Author

Vaclav Kotesovec, Oct 17 2016

Keywords

Crossrefs

Programs

  • Mathematica
    CoefficientList[Series[-Exp[2*x]*LambertW[-x], {x, 0, 20}], x]*Range[0, 20]!
    Table[Sum[Binomial[n, m]*Sum[Binomial[m, k]*k^(k-1), {k, 1, m}], {m, 1, n}], {n, 0, 20}]
  • PARI
    x='x+O('x^50); concat([0], Vec(serlaplace(- exp(2*x)*lambertw(-x) ))) \\ G. C. Greubel, Nov 08 2017

Formula

a(n) = Sum_{m=1..n} (binomial(n,m) * Sum_{k=1..m} binomial(m,k)*k^(k-1)).
a(n) ~ exp(2*exp(-1)) * n^(n-1).

A290215 a(n) = n! * [x^n] -exp(-n*x)*LambertW(-x).

Original entry on oeis.org

0, 1, -2, 18, -144, 1900, -26820, 485394, -9679936, 225394488, -5765768100, 164923889350, -5132384691984, 174433050454260, -6385752833589220, 251596880714336850, -10585338808667808000, 474507594337155230704, -22550580127644413987268
Offset: 0

Views

Author

Ilya Gutkovskiy, Oct 06 2017

Keywords

Comments

The n-th term of the n-th inverse binomial transform of A000169 (with A000169(0) = 0).

Crossrefs

Programs

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

Formula

a(n) ~ -(-1)^n * LambertW(1) * n^n. - Vaclav Kotesovec, Oct 06 2017
Showing 1-3 of 3 results.