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 1-10 of 13 results. Next

A371121 E.g.f. satisfies A(x) = 1 - x*A(x)*log(1 - x*A(x)).

Original entry on oeis.org

1, 0, 2, 3, 56, 330, 5724, 68460, 1351552, 24594192, 578257200, 13915923120, 389216689344, 11518744311360, 377576873670528, 13185760854520800, 497969104450867200, 19992393239486976000, 856421361373185137664, 38819358713756193292800
Offset: 0

Views

Author

Seiichi Manyama, Mar 11 2024

Keywords

Links

Crossrefs

Cf. A370993, A371117, A371122.
Cf. A371119.

Programs

  • PARI
    a(n) = n!^2*sum(k=0, n\2, abs(stirling(n-k, k, 1))/((n-k)!*(n-k+1)!));

Formula

a(n) = (n!)^2 * Sum_{k=0..floor(n/2)} |Stirling1(n-k,k)|/( (n-k)! * (n-k+1)! ).
E.g.f.: (1/x) * Series_Reversion( x/(1 - x*log(1 - x)) ). - Seiichi Manyama, Sep 19 2024

A371230 E.g.f. satisfies A(x) = 1 - x*A(x)^3*log(1 - x*A(x)^2).

Original entry on oeis.org

1, 0, 2, 3, 128, 750, 29964, 377160, 15795072, 329631120, 15001287120, 449174341440, 22551082739712, 885381886509120, 49302509206648320, 2391802812599316480, 147728974730632012800, 8502972330919072688640, 580806950108814502345728
Offset: 0

Views

Author

Seiichi Manyama, Mar 15 2024

Keywords

Crossrefs

Cf. A371227, A371228, A371229.
Cf. A370993, A371232.
Cf. A371122.

Programs

  • Mathematica
    terms=19; A[]=1; Do[A[x]= 1 - x*A[x]^3*Log[1 - x*A[x]^2] + O[x]^terms//Normal, terms]; CoefficientList[Series[A[x],{x,0,terms}],x]*Range[0,terms-1]! (* Stefano Spezia, Sep 03 2025 *)
  • PARI
    a(n) = n!*sum(k=0, n\2, (2*n+k)!*abs(stirling(n-k, k, 1))/(n-k)!)/(2*n+1)!;

Formula

a(n) = (n!/(2*n+1)!) * Sum_{k=0..floor(n/2)} (2*n+k)! * |Stirling1(n-k,k)|/(n-k)!.

A371229 E.g.f. satisfies A(x) = 1 - x*A(x)^2*log(1 - x*A(x)^2).

Original entry on oeis.org

1, 0, 2, 3, 104, 630, 19944, 259560, 8718464, 185086944, 6914815200, 206059083120, 8700740615808, 332779651158240, 15916427365716864, 738672634596405600, 39847940942657495040, 2163098542598925281280, 130682368989193123952640
Offset: 0

Views

Author

Seiichi Manyama, Mar 15 2024

Keywords

Crossrefs

Cf. A371227, A371228, A371230.
Cf. A371121, A371231.
Cf. A370993.

Programs

  • PARI
    a(n) = n!*(2*n)!*sum(k=0, n\2, abs(stirling(n-k, k, 1))/((n-k)!*(2*n-k+1)!));

Formula

a(n) = n! * (2*n)! * Sum_{k=0..floor(n/2)} |Stirling1(n-k,k)|/( (n-k)! * (2*n-k+1)! ).

A370994 Expansion of e.g.f. (1/x) * Series_Reversion( x*(1 + x^2*log(1-x)) ).

Original entry on oeis.org

1, 0, 0, 6, 12, 40, 3060, 23688, 191520, 9698400, 158548320, 2304973440, 100716073920, 2627516361600, 58513944513024, 2512156283683200, 89046056086041600, 2739316757454950400, 124170651534918297600, 5440968468533003212800, 215067442349096186572800
Offset: 0

Views

Author

Seiichi Manyama, Mar 06 2024

Keywords

Links

Crossrefs

Cf. A052802, A370993, A370995.
Cf. A351503, A370989.

Programs

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

Formula

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

A371232 E.g.f. satisfies A(x) = 1 - x*A(x)^4*log(1 - x*A(x)^3).

Original entry on oeis.org

1, 0, 2, 3, 176, 1050, 57144, 744660, 41682304, 917959392, 54654865920, 1761420386880, 113338947830976, 4879197834619680, 341937322823859840, 18486700938579444480, 1415296984669095859200, 92017658919053166405120, 7695907229874069158658048
Offset: 0

Views

Author

Seiichi Manyama, Mar 15 2024

Keywords

Crossrefs

Cf. A370993, A371230.

Programs

  • PARI
    a(n) = n!*sum(k=0, n\2, (3*n+k)!*abs(stirling(n-k, k, 1))/(n-k)!)/(3*n+1)!;

Formula

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

A371122 E.g.f. satisfies A(x) = 1 - x*A(x)^3*log(1 - x*A(x)).

Original entry on oeis.org

1, 0, 2, 3, 104, 570, 19284, 220500, 7975008, 148889664, 5911249680, 157016471040, 6913129099392, 239681708117280, 11734594390915200, 501510627153244800, 27265653826293749760, 1380895751066249779200, 83060557136719693406208
Offset: 0

Views

Author

Seiichi Manyama, Mar 11 2024

Keywords

Crossrefs

Cf. A370993, A371117, A371121.
Cf. A371120.

Programs

  • PARI
    a(n) = n!*sum(k=0, n\2, (n+2*k)!*abs(stirling(n-k, k, 1))/((n-k)!*(n+k+1)!));

Formula

a(n) = n! * Sum_{k=0..floor(n/2)} (n+2*k)! * |Stirling1(n-k,k)|/( (n-k)! * (n+k+1)! ).

A376344 Expansion of e.g.f. (1/x) * Series_Reversion( x*(1 + x*log(1-x^2)) ).

Original entry on oeis.org

1, 0, 0, 6, 0, 60, 2880, 1680, 201600, 8074080, 19958400, 1824197760, 69854400000, 436929292800, 36099561738240, 1392369634656000, 17026966410854400, 1344523178718720000, 54023115000830976000, 1095484919871908966400, 84994409643640713216000, 3650011125774294048768000, 109122812080533877712486400
Offset: 0

Views

Author

Seiichi Manyama, Sep 21 2024

Keywords

Links

Crossrefs

Cf. A370993, A376346.
Cf. A370994, A375561.

Programs

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

Formula

a(n) = (1/(n+1)) * Sum_{k=0..floor(n/2)} (2*n-2*k)! * |Stirling1(k,n-2*k)|/k!.

A376386 Expansion of e.g.f. (1/x) * Series_Reversion( x*(1 + x*log(1-x))^3 ).

Original entry on oeis.org

1, 0, 6, 9, 600, 3510, 204372, 2617020, 152727936, 3319236144, 203151929040, 6485780434320, 425284393933440, 18190896271479360, 1291781802823916544, 69545182272420909600, 5374429456543444177920, 348502600060029871948800, 29344904433432469953368064
Offset: 0

Views

Author

Seiichi Manyama, Sep 22 2024

Keywords

Links

Crossrefs

Cf. A370993, A376385.
Cf. A371232, A376387.
Cf. A375672.

Programs

  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(serreverse(x*(1+x*log(1-x))^3)/x))
  • PARI
    a(n) = 3*n!*sum(k=0, n\2, (3*n+k+2)!*abs(stirling(n-k, k, 1))/(n-k)!)/(3*n+3)!;

Formula

E.g.f. A(x) satisfies A(x) = 1/(1 + x*A(x) * log(1 - x*A(x)))^3.
E.g.f.: B(x)^3, where B(x) is the e.g.f. of A371232.
a(n) = (3 * n!/(3*n+3)!) * Sum_{k=0..floor(n/2)} (3*n+k+2)! * |Stirling1(n-k,k)|/(n-k)!.

A370995 Expansion of e.g.f. (1/x) * Series_Reversion( x*(1 + x^3*log(1-x)) ).

Original entry on oeis.org

1, 0, 0, 0, 24, 60, 240, 1260, 209664, 2056320, 20476800, 221205600, 19370292480, 406935809280, 7376151444480, 131868581644800, 8376837844193280, 282378273124147200, 7891890567682867200, 207283550601631795200, 11520967360247698636800
Offset: 0

Views

Author

Seiichi Manyama, Mar 06 2024

Keywords

Links

Crossrefs

Cf. A052802, A370993, A370994.
Cf. A351504.

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(serreverse(x*(1+x^3*log(1-x)))/x))
  • PARI
    a(n) = sum(k=0, n\4, (n+k)!*abs(stirling(n-3*k, k, 1))/(n-3*k)!)/(n+1);

Formula

a(n) = (1/(n+1)) * Sum_{k=0..floor(n/4)} (n+k)! * |Stirling1(n-3*k,k)|/(n-3*k)!.

A376385 Expansion of e.g.f. (1/x) * Series_Reversion( x*(1 + x*log(1-x))^2 ).

Original entry on oeis.org

1, 0, 4, 6, 280, 1620, 67788, 844200, 36344992, 752867136, 34869857040, 1039132179360, 52776841318848, 2066262237673920, 115959403155851136, 5617102749187849920, 348802585405252070400, 20063354348482794961920, 1375625132090917881338880
Offset: 0

Views

Author

Seiichi Manyama, Sep 22 2024

Keywords

Links

Crossrefs

Cf. A370993, A376386.
Cf. A371230, A375671.

Programs

  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(serreverse(x*(1+x*log(1-x))^2)/x))
  • PARI
    a(n) = 2*n!*sum(k=0, n\2, (2*n+k+1)!*abs(stirling(n-k, k, 1))/(n-k)!)/(2*n+2)!;

Formula

E.g.f. A(x) satisfies A(x) = 1/(1 + x*A(x) * log(1 - x*A(x)))^2.
E.g.f.: B(x)^2, where B(x) is the e.g.f. of A371230.
a(n) = (2 * n!/(2*n+2)!) * Sum_{k=0..floor(n/2)} (2*n+k+1)! * |Stirling1(n-k,k)|/(n-k)!.
Showing 1-10 of 13 results. Next

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