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 1-7 of 7 results.

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

A371115 E.g.f. satisfies A(x) = 1 + x*(exp(x*A(x)) - 1).

Original entry on oeis.org

1, 0, 2, 3, 28, 185, 1566, 18277, 218744, 3206961, 52134490, 935303501, 18733723812, 406458491881, 9598660337462, 244471271572725, 6671672053304176, 194631575264393057, 6036199529439919410, 198427339307102272669, 6892068588221322730460
Offset: 0

Views

Author

Seiichi Manyama, Mar 11 2024

Keywords

Crossrefs

Cf. A000272, A371116.
Cf. A371117.

Programs

  • Mathematica
    nmax = 20; CoefficientList[Series[1 - x - ProductLog[-E^((1 - x)*x)*x^2]/x, {x, 0, nmax}], x] * Range[0, nmax]! (* Vaclav Kotesovec, Mar 11 2024 *)
  • PARI
    a(n) = n!*sum(k=0, n\2, stirling(n-k, k, 2)/(n-2*k+1)!);

Formula

a(n) = n! * Sum_{k=0..floor(n/2)} Stirling2(n-k,k)/(n-2*k+1)!.
From Vaclav Kotesovec, Mar 11 2024: (Start)
E.g.f.: 1 - x - LambertW(-exp((1 - x)*x)*x^2)/x.
a(n) ~ sqrt(2 + r - 2*r^2) * n^(n-1) / (exp(n) * r^(n+1)), where r = 0.5356007344755967412570670018666980389185523835846... if the root of the equation exp(1 + r - r^2) * r^2 = 1. (End)

A371227 E.g.f. satisfies A(x) = 1 - x*log(1 - x*A(x)^2).

Original entry on oeis.org

1, 0, 2, 3, 56, 390, 6384, 92400, 1812768, 38565072, 949927680, 25934040000, 783458550720, 25909868761920, 930720395219328, 36108805836317760, 1504050682102456320, 66964478742976711680, 3173178938051223889920, 159461567895099436047360
Offset: 0

Views

Author

Seiichi Manyama, Mar 15 2024

Keywords

Crossrefs

Cf. A371228, A371229, A371230.
Cf. A371117.

Programs

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

Formula

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

A371118 E.g.f. satisfies A(x) = 1 - x^2*log(1 - x*A(x)).

Original entry on oeis.org

1, 0, 0, 6, 12, 40, 900, 8568, 80640, 1170720, 19625760, 335079360, 6259524480, 132897697920, 3078950434560, 75934287475200, 2010354982272000, 57241482249369600, 1735591000023336960, 55694476130213652480, 1889613850762113638400
Offset: 0

Views

Author

Seiichi Manyama, Mar 11 2024

Keywords

Crossrefs

Cf. A138013, A371117.
Cf. A371116.

Programs

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

Formula

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

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)! ).

A377685 E.g.f. satisfies A(x) = (1 - x * log(1 - x*A(x)))^2.

Original entry on oeis.org

1, 0, 4, 6, 136, 900, 16308, 229320, 4691104, 99156960, 2481162480, 67862678400, 2063842827264, 68473763804160, 2468786906210688, 96048626176339200, 4010912604492410880, 178968539487145282560, 8496991445958129576960, 427734144995749047152640
Offset: 0

Views

Author

Seiichi Manyama, Nov 04 2024

Keywords

Crossrefs

Cf. A371117, A377686.
Cf. A371227, A377390.

Programs

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

Formula

E.g.f.: B(x)^2, where B(x) is the e.g.f. of A371227.
a(n) = 2 * n! * Sum_{k=0..floor(n/2)} (2*n-2*k+1)! * |Stirling1(n-k,k)|/( (n-k)! * (2*n-3*k+2)! ).

A377686 E.g.f. satisfies A(x) = (1 - x * log(1 - x*A(x)))^3.

Original entry on oeis.org

1, 0, 6, 9, 312, 2070, 53892, 797580, 21541440, 508313232, 15840608400, 502075577520, 18473543511552, 722232734446080, 31135359390952320, 1435933667363963040, 71392285554374384640, 3782802775152784320000, 213512536856209839796224, 12767785967296083820561920
Offset: 0

Views

Author

Seiichi Manyama, Nov 04 2024

Keywords

Crossrefs

Cf. A371117, A377685.
Cf. A377391, A377687.

Programs

  • PARI
    a(n) = 3*n!*sum(k=0, n\2, (3*n-3*k+2)!*abs(stirling(n-k, k, 1))/((n-k)!*(3*n-4*k+3)!));

Formula

E.g.f.: B(x)^3, where B(x) is the e.g.f. of A377687.
a(n) = 3 * n! * Sum_{k=0..floor(n/2)} (3*n-3*k+2)! * |Stirling1(n-k,k)|/( (n-k)! * (3*n-4*k+3)! ).
Showing 1-7 of 7 results.

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