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.

A305819 Expansion of e.g.f. 1/(1 + LambertW(-log(1 + x))).

Original entry on oeis.org

1, 1, 3, 17, 132, 1334, 16442, 239994, 4041776, 77183328, 1647541632, 38877352392, 1004869488048, 28234217634024, 856830099396840, 27930093941047464, 973269467390922240, 36104568839480990400, 1420556927968241858880, 59088101641333114906944, 2590680379402887359111424
Offset: 0

Views

Author

Ilya Gutkovskiy, Aug 18 2018

Keywords

Comments

Inverse Stirling transform of A000312.

Links

Crossrefs

Cf. A000312, A052807, A277489, A282190, A305981.

Programs

  • Maple
    a:=series(1/(1+LambertW(-log(1+x))),x=0,21): seq(n!*coeff(a,x,n),n=0..20); # Paolo P. Lava, Mar 26 2019
  • Mathematica
    nmax = 20; CoefficientList[Series[1/(1 + LambertW[-Log[1 + x]]), {x, 0, nmax}], x] Range[0, nmax]!
Join[{1}, Table[Sum[StirlingS1[n, k] k^k, {k, n}], {n, 20}]]
  • PARI
    a(n) = sum(k=0, n, k^k*stirling(n, k, 1)); \\ Seiichi Manyama, Feb 05 2022
  • PARI
    my(N=40, x='x+O('x^N)); Vec(serlaplace(1/(1+lambertw(-log(1+x))))) \\ Seiichi Manyama, Feb 05 2022

Formula

a(n) = Sum_{k=0..n} Stirling1(n,k)*k^k.
a(n) ~ n^n / ((exp(exp(-1)) - 1)^(n + 1/2) * exp(n + (1 - exp(-1))/2)). - Vaclav Kotesovec, Aug 18 2018

A351218 a(n) = Sum_{k=0..n} (-k)^k * Stirling2(n,k).

Original entry on oeis.org

1, -1, 3, -16, 121, -1181, 14114, -199543, 3257139, -60279214, 1247164055, -28525394481, 714681439212, -19465007759913, 572609747089735, -18093710202583480, 611202186074834221, -21979340746682042249, 838330656532184312218, -33803668628843391999843
Offset: 0

Views

Author

Seiichi Manyama, Feb 05 2022

Keywords

Links

Crossrefs

Cf. A282190, A305981.

Programs

  • Maple
    b:= proc(n, m) option remember; `if`(n=0,
      (-m)^m, m*b(n-1, m)+b(n-1, m+1))
    end:
a:= n-> b(n, 0):
seq(a(n), n=0..20);  # Alois P. Heinz, Jul 17 2022
  • Mathematica
    Table[Sum[(-1)^k * k^k * StirlingS2[n,k], {k,1,n}], {n,0,20}] (* Vaclav Kotesovec, Feb 05 2022 *)
  • PARI
    a(n) = sum(k=0, n, (-k)^k*stirling(n, k, 2));
  • PARI
    my(N=40, x='x+O('x^N)); Vec(serlaplace(1/(1+lambertw(exp(x)-1))))

Formula

E.g.f.: 1/(1 + LambertW(exp(x) - 1)), where LambertW() is the Lambert W-function.
a(n) ~ (-1)^n * n^n / (sqrt(exp(1)-1) * (1 - log(exp(1)-1))^(n + 1/2) * exp(n)). - Vaclav Kotesovec, Feb 05 2022

A362891 Expansion of e.g.f. 1/(1 + LambertW(x^2 * log(1-x))).

Original entry on oeis.org

1, 0, 0, 6, 12, 40, 1620, 11088, 80640, 2289600, 30471840, 374663520, 9819817920, 195106129920, 3507260492736, 95860364846400, 2466492401318400, 58909563259223040, 1775000008437557760, 54856736708999339520, 1629826915777548364800
Offset: 0

Views

Author

Seiichi Manyama, May 08 2023

Keywords

Links

Crossrefs

Cf. A305981, A362835.
Cf. A353228, A362892.

Programs

  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(1/(1+lambertw(x^2*log(1-x)))))

Formula

a(n) = n! * Sum_{k=0..floor(n/3)} k^k * |Stirling1(n-2*k,k)|/(n-2*k)!.
Showing 1-3 of 3 results.

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