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.

A377360 E.g.f. satisfies A(x) = ( 1 - log(1 - x*A(x)) )^2.

Original entry on oeis.org

1, 2, 12, 130, 2082, 44488, 1192964, 38557860, 1459988440, 63414711072, 3108861424032, 169829819311392, 10230860299538400, 673850170929176928, 48176129912775680160, 3715759452364764485280, 307545698210584533055488, 27190399275422185989742080, 2557448587458299889542868480
Offset: 0

Views

Author

Seiichi Manyama, Oct 26 2024

Keywords

Links

Crossrefs

Cf. A138013, A377361.
Cf. A097629, A367080, A376392.

Programs

  • Mathematica
    nmax = 20; CoefficientList[1/x * InverseSeries[Series[x/(1 - Log[1 - x])^2, {x, 0, nmax + 1}], x], x] * Range[0, nmax]! (* Vaclav Kotesovec, Aug 27 2025 *)
  • PARI
    a(n) = 2*(2*n+1)!*sum(k=0, n, abs(stirling(n, k, 1))/(2*n-k+2)!);

Formula

E.g.f.: B(x)^2, where B(x) is the e.g.f. of A367080.
a(n) = 2 * (2*n+1)! * Sum_{k=0..n} |Stirling1(n,k)|/(2*n-k+2)!.
E.g.f.: (1/x) * Series_Reversion( x/(1 - log(1-x))^2 ).
a(n) ~ sqrt(2) * LambertW(-1, -2*exp(-3))^n * (2 + LambertW(-1, -2*exp(-3)))^(n+2) * n^(n-1) / (exp(n) * sqrt(-1 - LambertW(-1, -2*exp(-3)))). - Vaclav Kotesovec, Aug 27 2025

A377427 E.g.f. satisfies A(x) = 1/(1 + log(1 - x*A(x)^2))^2.

Original entry on oeis.org

1, 2, 24, 574, 20950, 1034588, 64592556, 4881978904, 433485612000, 44236604978112, 5102049359506176, 656355318561027072, 93184708368255490896, 14472905373087118415040, 2441090221004851173202080, 444344375119629711627403776, 86822659466273927313499224192
Offset: 0

Views

Author

Seiichi Manyama, Oct 28 2024

Keywords

Crossrefs

Cf. A377426, A377429.
Cf. A376392.

Programs

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

Formula

E.g.f.: B(x)^2, where B(x) is the e.g.f. of A377426.
a(n) = (2/(4*n+2)!) * Sum_{k=0..n} (4*n+k+1)! * |Stirling1(n,k)|.

A377429 Expansion of e.g.f. (1/x) * Series_Reversion( x*(1 + log(1-x))^4 ).

Original entry on oeis.org

1, 4, 56, 1436, 54540, 2763696, 175688744, 13457185080, 1207241712536, 124205544781728, 14420516981211360, 1865347268407271040, 266056506383725529568, 41485848013549310521536, 7021170794004780911946048, 1281852242007649764308226240, 251124461130948243588667169280
Offset: 0

Views

Author

Seiichi Manyama, Oct 28 2024

Keywords

Links

Crossrefs

Cf. A052802, A376392, A376393.
Cf. A377426, A377427.

Programs

  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(serreverse(x*(1+log(1-x))^4)/x))
  • PARI
    a(n) = 4*sum(k=0, n, (4*n+k+3)!*abs(stirling(n, k, 1)))/(4*n+4)!;

Formula

E.g.f. A(x) satisfies A(x) = 1/(1 + log(1 - x*A(x)))^4.
E.g.f.: B(x)^4, where B(x) is the e.g.f. of A377426.
a(n) = (4/(4*n+4)!) * Sum_{k=0..n} (4*n+k+3)! * |Stirling1(n,k)|.

A376436 Expansion of e.g.f. (1/x) * Series_Reversion( x*(1 + x^2*log(1-x))^2 ).

Original entry on oeis.org

1, 0, 0, 12, 24, 80, 11160, 87696, 715680, 62337600, 1065980160, 15842534400, 1109943362880, 31591940440320, 731706348941568, 46767587926752000, 1889337264901632000, 61735665488234250240, 3896148715287564902400, 201584132714100384460800, 8661099107269708639948800, 567405718655558932535500800
Offset: 0

Views

Author

Seiichi Manyama, Sep 22 2024

Keywords

Links

Crossrefs

Cf. A376385, A376392.
Cf. A371235, A375639.

Programs

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

Formula

E.g.f. A(x) satisfies A(x) = 1/(1 + x^2*A(x)^2 * log(1 - x*A(x)))^2.
E.g.f.: B(x)^2, where B(x) is the e.g.f. of A371235.
a(n) = (2 * n!/(2*n+2)!) * Sum_{k=0..floor(n/3)} (2*n+k+1)! * |Stirling1(n-2*k,k)|/(n-2*k)!.
Showing 1-4 of 4 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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