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 1-6 of 6 results.

A052820 Expansion of e.g.f. 1/(1 - x + log(1 - x)).

Original entry on oeis.org

1, 2, 9, 62, 572, 6604, 91526, 1480044, 27353448, 568731648, 13138994112, 333895239072, 9256507508112, 278000959058016, 8991458660924112, 311585506208924064, 11517363473843526912, 452332548042633835776
Offset: 0

Views

Author

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

Keywords

Comments

Previous name was: A simple grammar.
a(n) is the number of ways to seat n people at circular tables, then linearly order the tables, then designate some (possibly all or none) of the tables at which only one person is seated. a(2) = 9 because we have: (1)(2), (1')(2), (1)(2'), (1')(2'), (2)(1), (2')(1), (2)(1'), (2')(1'), (1,2). Cf. A007840. - Geoffrey Critzer, Nov 05 2013

Crossrefs

Programs

  • Maple
    spec := [S,{C=Cycle(Z),B=Union(C,Z),S=Sequence(B)},labeled]: seq(combstruct[count](spec,size=n), n=0..20);
  • Mathematica
    CoefficientList[Series[1/(1-x+Log[1-x]), {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Oct 01 2013 *)

Formula

E.g.f.: -1/(-1+x+log(-1/(-1+x))).
a(n) ~ n! * (1/(1-LambertW(1)))^n/(1/LambertW(1)-LambertW(1)). - Vaclav Kotesovec, Oct 01 2013
a(0) = 1; a(n) = n * a(n-1) + Sum_{k=0..n-1} binomial(n,k) * (n-k-1)! * a(k). - Ilya Gutkovskiy, Apr 26 2021

Extensions

New name using e.g.f., Vaclav Kotesovec, Oct 01 2013

A367846 Expansion of e.g.f. 1/(1 - x + log(1 - 3*x)).

Original entry on oeis.org

1, 4, 41, 654, 14028, 377112, 12177126, 458916588, 19769059944, 958125646080, 51597765220608, 3056601306206016, 197532472461453072, 13829353660386169344, 1042679226974498229456, 84229294995413626072608, 7257792124889497549663488
Offset: 0

Views

Author

Seiichi Manyama, Dec 02 2023

Keywords

Crossrefs

Programs

  • PARI
    a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=i*v[i]+sum(j=1, i, 3^j*(j-1)!*binomial(i, j)*v[i-j+1])); v;

Formula

a(0) = 1; a(n) = n * a(n-1) + Sum_{k=1..n} 3^k * (k-1)! * binomial(n,k) * a(n-k).

A367847 Expansion of e.g.f. 1/(1 - x + log(1 - 4*x)).

Original entry on oeis.org

1, 5, 66, 1358, 37592, 1304536, 54384080, 2646247152, 147186205056, 9210766696320, 640472632680192, 48989958019395840, 4087959251421060096, 369547591764702870528, 35976590549993421907968, 3752609987262290143082496, 417518648351593243448279040
Offset: 0

Views

Author

Seiichi Manyama, Dec 02 2023

Keywords

Crossrefs

Programs

  • PARI
    a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=i*v[i]+sum(j=1, i, 4^j*(j-1)!*binomial(i, j)*v[i-j+1])); v;

Formula

a(0) = 1; a(n) = n * a(n-1) + Sum_{k=1..n} 4^k * (k-1)! * binomial(n,k) * a(n-k).

A367851 Expansion of e.g.f. 1/(1 - x + log(1 - 2*x)/2).

Original entry on oeis.org

1, 2, 10, 80, 872, 11984, 198416, 3840192, 85031040, 2119385856, 58714881792, 1789646610432, 59515302478848, 2144299161348096, 83204666280609792, 3459286210445942784, 153413140701637804032, 7228914528043587796992, 360670654712328998289408
Offset: 0

Views

Author

Seiichi Manyama, Dec 02 2023

Keywords

Crossrefs

Programs

  • PARI
    a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=i*v[i]+sum(j=1, i, 2^(j-1)*(j-1)!*binomial(i, j)*v[i-j+1])); v;

Formula

a(0) = 1; a(n) = n * a(n-1) + Sum_{k=1..n} 2^(k-1) * (k-1)! * binomial(n,k) * a(n-k).

A367828 E.g.f. A(x) satisfies A(x) = (1 - log(1 - x) * A(2*x)) / (1 - x).

Original entry on oeis.org

1, 2, 13, 209, 7874, 687194, 138026428, 63273019396, 65547617642192, 151904702763916944, 780028188748068778464, 8799101018162158392857376, 216405047530763040469557821568, 11527355297347542160143184818391680, 1322291382391922104463259686181056293632
Offset: 0

Views

Author

Seiichi Manyama, Dec 02 2023

Keywords

Crossrefs

Programs

  • PARI
    a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=i*v[i]+sum(j=1, i, 2^(i-j)*(j-1)!*binomial(i, j)*v[i-j+1])); v;

Formula

a(0) = 1; a(n) = n * a(n-1) + Sum_{k=1..n} 2^(n-k) * (k-1)! * binomial(n,k) * a(n-k).

A367922 Expansion of e.g.f. 1/(1 - x + 2*log(1 - x)).

Original entry on oeis.org

1, 3, 20, 202, 2724, 45928, 929288, 21936864, 591822912, 17962293408, 605744322048, 22470338716032, 909323575700352, 39864781715364864, 1882110048700328448, 95205899353680970752, 5137022051563160623104, 294501790029090740576256
Offset: 0

Views

Author

Seiichi Manyama, Dec 05 2023

Keywords

Crossrefs

Programs

  • PARI
    a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=i*v[i]+2*sum(j=1, i, (j-1)!*binomial(i, j)*v[i-j+1])); v;

Formula

a(0) = 1; a(n) = n * a(n-1) + 2 * Sum_{k=1..n} (k-1)! * binomial(n,k) * a(n-k).
Showing 1-6 of 6 results.