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 1-4 of 4 results.

A362244 Expansion of e.g.f. 1/(1 - x * exp(-x * (exp(-x) - 1))).

Original entry on oeis.org

1, 1, 2, 12, 60, 440, 3810, 37212, 430696, 5482080, 78252390, 1227201140, 20955546348, 388492703040, 7745445183658, 165550236166980, 3773990094033360, 91401848785134272, 2344168680183033678, 63455096201600595060, 1808160553359068792020
Offset: 0

Views

Author

Seiichi Manyama, Apr 12 2023

Keywords

Crossrefs

Cf. A362245, A362246, A362247.
Cf. A362237.

Programs

  • Mathematica
    With[{nn=20},CoefficientList[Series[1/(1-x Exp[-x(Exp[-x]-1)]),{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Dec 27 2024 *)
  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(1/(1-x*exp(-x*(exp(-x)-1)))))

Formula

a(n) = n! * Sum_{i=0..n} (-1)^(n-i) * ( Sum_{j=0..n-i} i^j * Stirling2(n-i-j,j)/(n-i-j)! ).

A362245 Expansion of e.g.f. 1/(1 - x * exp(x * (exp(x) - 1))).

Original entry on oeis.org

1, 1, 2, 12, 84, 680, 6750, 78372, 1035608, 15402816, 254672730, 4631221100, 91872810612, 1974481960464, 45698618329910, 1133221107064620, 29974735063385520, 842413032202481792, 25067919890384214066, 787394937539847359052, 26034146454319615550540
Offset: 0

Views

Author

Seiichi Manyama, Apr 12 2023

Keywords

Crossrefs

Cf. A362244, A362246, A362247.
Cf. A362238.

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(1/(1-x*exp(x*(exp(x)-1)))))

Formula

a(n) = n! * Sum_{i=0..n} Sum_{j=0..n-i} i^j * Stirling2(n-i-j,j)/(n-i-j)!.
a(n) ~ n! / ((1 - r + exp(r)*r*(1 + r)) * r^n), where r = 0.60489399462026660841486230237937164068755854932856922096976397761... is the root of the equation exp(r*(exp(r)-1)) = 1/r. - Vaclav Kotesovec, Apr 13 2023

A362246 Expansion of e.g.f. exp(x * exp(-x * (exp(-x) - 1))).

Original entry on oeis.org

1, 1, 1, 7, 13, 81, 391, 1093, 18537, 577, 1013131, -54339, 39429853, 424162753, -4130181873, 137131757701, -1733851435439, 29708533549953, -337960798083053, 3890007865959661, -11844138798049659, -662440203084569279, 30135319297423429783
Offset: 0

Views

Author

Seiichi Manyama, Apr 12 2023

Keywords

Crossrefs

Cf. A362244, A362245, A362247.
Cf. A347726.

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(x*exp(-x*(exp(-x)-1)))))

Formula

a(n) = n! * Sum_{i=0..n} (-1)^(n-i) * ( Sum_{j=0..n-i} i^j * Stirling2(n-i-j,j)/(n-i-j)! )/i!.

A362275 Expansion of e.g.f. exp(x*exp(-x*exp(-x))).

Original entry on oeis.org

1, 1, -1, 4, -3, -44, 595, -5214, 36953, -176840, -498969, 31239550, -605916971, 9001842708, -110410388101, 998140500826, -1114748107215, -247879285927184, 8410011463871695, -201620611378992906, 4024515940445770541, -66832538504513547380
Offset: 0

Views

Author

Seiichi Manyama, Apr 13 2023

Keywords

Links

Crossrefs

Cf. A000949, A362273, A362274.
Cf. A362247.

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(x*exp(-x*exp(-x)))))

Formula

a(n) = Sum_{i=0..n} (-1)^(n-i) * binomial(n,i) * Sum_{j=0..n-i} i^j * j^(n-i-j) * binomial(n-i,j).
Showing 1-4 of 4 results.

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