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 1-5 of 5 results.

A052801 A simple grammar: labeled pairs of sequences of cycles.

Original entry on oeis.org

1, 2, 8, 46, 342, 3108, 33324, 411360, 5741856, 89379120, 1534623936, 28804923024, 586686138384, 12885385945248, 303537419684064, 7633673997722496, 204125888803996800, 5782960189212871680
Offset: 0

Views

Author

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

Keywords

Links

Crossrefs

Cf. A215916.
Cf. A007840, A354122, A354123.

Programs

  • Maple
    spec := [S,{C=Cycle(Z),B=Sequence(C),S=Prod(B,B)},labeled]: seq(combstruct[count](spec,size=n), n=0..20);
  • Mathematica
    CoefficientList[Series[1/(1+Log[1-x])^2, {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Sep 30 2013 *)
  • Maxima
    makelist(sum((-1)^(n-k)*stirling1(n, k)*(k+1)!, k, 0, n), n, 0, 17); /* Bruno Berselli, May 25 2011 */

Formula

E.g.f.: 1/(-1+log(-1/(-1+x)))^2.
a(n) = Sum_{k=0..n} (-1)^(n-k)*Stirling1(n, k)*(k+1)!. - Vladeta Jovovic, Sep 21 2003
a(n) = D^n(1/(1-x)^2) evaluated at x = 0, where D is the operator exp(x)*d/dx. Cf. A052811. - Peter Bala, Nov 25 2011
a(n) ~ n! * n*exp(n)/(exp(1)-1)^(n+2). - Vaclav Kotesovec, Sep 30 2013
From Anton Zakharov, Aug 07 2016: (Start)
a(n) = A007840(n) + A215916(n).
a(n) = Sum_{k=2..n+1} k!*s(n,k) where s(n,k) is the unsigned Stirling number of the first kind, (A132393). (End)
a(0) = 1; a(n) = Sum_{k=1..n} (k/n + 1) * (k-1)! * binomial(n,k) * a(n-k). - Seiichi Manyama, Nov 19 2023

A354122 Expansion of e.g.f. 1/(1 + log(1 - x))^3.

Original entry on oeis.org

1, 3, 15, 102, 870, 8892, 105708, 1431168, 21722136, 365105928, 6729341832, 134915992560, 2922576142320, 68013701197920, 1692075061072800, 44810389419079680, 1258472984174461440, 37357062009383877120, 1168635883239630120960, 38424619272539153157120
Offset: 0

Views

Author

Seiichi Manyama, May 17 2022

Keywords

Crossrefs

Cf. A007840, A052801, A354123.
Cf. A226515, A354120.

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(1/(1+log(1-x))^3))
  • PARI
    a(n) = sum(k=0, n, (k+2)!*abs(stirling(n, k, 1)))/2;

Formula

a(n) = (1/2) * Sum_{k=0..n} (k + 2)! * |Stirling1(n,k)|.
a(n) ~ sqrt(Pi/2) * n^(n + 5/2) / (exp(1) - 1)^(n+3). - Vaclav Kotesovec, Jun 04 2022
a(0) = 1; a(n) = Sum_{k=1..n} (2*k/n + 1) * (k-1)! * binomial(n,k) * a(n-k). - Seiichi Manyama, Nov 19 2023

A354121 Expansion of e.g.f. 1/(1 - log(1 + x))^4.

Original entry on oeis.org

1, 4, 16, 68, 316, 1616, 9080, 55800, 373080, 2699520, 21035040, 175708320, 1566916320, 14862171840, 149429426880, 1587766126080, 17779538050560, 209295747832320, 2583920845209600, 33389139008678400, 450642388471395840, 6342869733912760320
Offset: 0

Views

Author

Seiichi Manyama, May 17 2022

Keywords

Comments

a(46) is negative. - Vaclav Kotesovec, Jun 04 2022
It appears that a(n) is negative for even n >= 46. - Felix Fröhlich, Jun 04 2022

Links

Crossrefs

Cf. A006252, A317280, A354120.
Cf. A226738, A354123.

Programs

  • Mathematica
    Table[Sum[(k+3)! * StirlingS1[n,k], {k,0,n}]/6, {n,0,20}] (* Vaclav Kotesovec, Jun 04 2022 *)
  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(1/(1-log(1+x))^4))
  • PARI
    a(n) = sum(k=0, n, (k+3)!*stirling(n, k, 1))/6;

Formula

a(n) = (1/6) * Sum_{k=0..n} (k + 3)! * Stirling1(n,k).
a(0) = 1; a(n) = Sum_{k=1..n} (-1)^(k-1) * (3 * k/n + 1) * (k-1)! * binomial(n,k) * a(n-k). - Seiichi Manyama, Nov 19 2023

A351739 Expansion of e.g.f. 1/(1 + log(1-x))^x.

Original entry on oeis.org

1, 0, 2, 6, 40, 295, 2688, 28588, 348864, 4802922, 73652110, 1245046836, 23003289912, 461188427544, 9972307487660, 231341792369010, 5731422576446208, 151032969213699536, 4218265874407103640, 124471244064061267032, 3869361472890037713560
Offset: 0

Views

Author

Seiichi Manyama, May 20 2022

Keywords

Crossrefs

Cf. A052801, A052809, A354122, A354123.

Programs

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

Formula

a(0) = 1; a(n) = Sum_{k=1..n} A052809(k) * binomial(n-1,k-1) * a(n-k).
a(n) ~ n! * exp(n) / (Gamma(1 - 1/exp(1)) * n^(1/exp(1)) * (exp(1) - 1)^(n + 1 - 1/exp(1))). - Vaclav Kotesovec, Jun 04 2022

A382830 a(n) = Sum_{k=0..n} binomial(n+k-1,k) * |Stirling1(n,k)| * k!.

Original entry on oeis.org

1, 1, 8, 102, 1804, 40890, 1131108, 36948240, 1391945616, 59411849040, 2833582748160, 149347596487056, 8620256620495584, 540775669746661440, 36636074309252234880, 2665704585421541790720, 207329122282259073044736, 17165075378189396045777280, 1507206260097615729874083840
Offset: 0

Views

Author

Ilya Gutkovskiy, Apr 06 2025

Keywords

Crossrefs

Cf. A007840, A052801, A277759, A305919, A354122, A354123.

Programs

  • Mathematica
    Table[Sum[Binomial[n + k - 1, k] Abs[StirlingS1[n, k]] k!, {k, 0, n}], {n, 0, 18}]
Table[n! SeriesCoefficient[1/(1 + Log[1 - x])^n, {x, 0, n}], {n, 0, 18}]

Formula

a(n) = n! * [x^n] 1 / (1 + log(1 - x))^n.
a(n) ~ LambertW(exp(2))^n * n^n / (sqrt(1 + LambertW(exp(2))) * exp(n) * (LambertW(exp(2)) - 1)^(2*n)). - Vaclav Kotesovec, Apr 06 2025
Showing 1-5 of 5 results.

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