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.

Previous Showing 11-17 of 17 results.

A355786 E.g.f. satisfies A(x) = 1/(1 - 2*x)^(A(x)/2).

Original entry on oeis.org

1, 1, 5, 42, 497, 7620, 143979, 3241406, 84847489, 2534788296, 85170416115, 3180919433802, 130771002469953, 5869920100483452, 285705285804636411, 14989889385040915830, 843420165009747027969, 50664760467069168337680, 3236433107379299238343779
Offset: 0

Views

Author

Seiichi Manyama, Jul 17 2022

Keywords

Crossrefs

Cf. A052813, A355779.

Programs

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

Formula

E.g.f.: exp( -LambertW(log(1-2*x)/2) ).
a(n) = Sum_{k=0..n} 2^(n-k) * (k+1)^(k-1) * |Stirling1(n,k)|.
From Vaclav Kotesovec, Jul 18 2022: (Start)
E.g.f.: 2*LambertW(log(1-2*x)/2) / log(1-2*x).
a(n) ~ 2^(n - 1/2) * n^(n-1) * exp(3/2 - n + 2*n*exp(-1)) / (exp(2*exp(-1)) - 1)^(n - 1/2). (End)

A356914 E.g.f. satisfies: A(x) = 1/(1 - x * A(x)^2)^A(x).

Original entry on oeis.org

1, 1, 8, 129, 3188, 106820, 4530174, 232744274, 14054237000, 975665799792, 76568358325440, 6703512043934112, 647772783078516600, 68484470168404178928, 7863452609057425825704, 974474039841523412062440, 129636661193226128284609344, 18427094124309886080021680640
Offset: 0

Views

Author

Seiichi Manyama, Sep 07 2022

Keywords

Crossrefs

Cf. A052813, A349556, A356960.
Cf. A355766, A356972.

Programs

  • Mathematica
    m = 18; (* number of terms *)
A[_] = 0;
Do[A[x_] = 1/(1 - x*A[x]^2)^A[x] + O[x]^m // Normal, {m}];
CoefficientList[A[x], x]*Range[0, m-1]! (* Jean-François Alcover, Sep 12 2022 *)
  • PARI
    a(n) = sum(k=0, n, (2*n+k+1)^(k-1)*abs(stirling(n, k, 1)));

Formula

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

A356927 E.g.f. satisfies A(x) = 1/(1 - x)^(A(x)/(1 - x)).

Original entry on oeis.org

1, 1, 6, 54, 676, 10980, 220488, 5289592, 147828896, 4721152320, 169723566240, 6785559484704, 298726260001728, 14362141350822720, 748845960914596608, 42092072779399215360, 2537464961261745635328, 163317885950059143238656
Offset: 0

Views

Author

Seiichi Manyama, Sep 04 2022

Keywords

Links

Crossrefs

Cf. A000254, A000272, A052813, A087761, A356925.

Programs

  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, (k+1)^(k-1)*(-log(1-x)/(1-x))^k/k!)))
  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(-lambertw(log(1-x)/(1-x)))))
  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace((1-x)*lambertw(log(1-x)/(1-x))/log(1-x)))

Formula

E.g.f.: A(x) = Sum_{k>=0} (k+1)^(k-1) * (-log(1-x)/(1-x))^k / k!.
E.g.f.: A(x) = exp( -LambertW(log(1-x)/(1-x)) ).
E.g.f.: A(x) = (1-x) * LambertW(log(1-x)/(1-x))/log(1-x).
a(n) ~ sqrt(1 + LambertW(exp(-1))) * n^(n-1) / (LambertW(exp(-1)) * exp(n - 1/2) * (1 - exp(1)*LambertW(exp(-1)))^(n - 1/2)). - Vaclav Kotesovec, Nov 14 2022

A357026 E.g.f. satisfies A(x) = (1 - x)^(log(1 - x) * A(x)).

Original entry on oeis.org

1, 0, 2, 6, 58, 460, 5528, 70308, 1098060, 18910512, 371480832, 8022952080, 191325228576, 4961955705408, 139572074260656, 4224646630879920, 137077496211066384, 4744151145076980864, 174517898073769832448, 6798949897214608689024, 279688643858492900930496
Offset: 0

Views

Author

Seiichi Manyama, Sep 09 2022

Keywords

Links

Crossrefs

Cf. A052813, A357027.

Programs

  • Mathematica
    m = 21; (* number of terms *)
A[_] = 0;
Do[A[x_] = (1 - x)^(Log[1 - x]*A[x]) + O[x]^m // Normal, {m}];
CoefficientList[A[x], x]*Range[0, m - 1]! (* Jean-François Alcover, Sep 12 2022 *)
  • PARI
    a(n) = sum(k=0, n\2, (2*k)!*(k+1)^(k-1)*abs(stirling(n, 2*k, 1))/k!);
  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, (k+1)^(k-1)*log(1-x)^(2*k)/k!)))
  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(-lambertw(-log(1-x)^2))))
  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(-lambertw(-log(1-x)^2)/log(1-x)^2))

Formula

E.g.f. satisfies log(A(x)) = log(1 - x)^2 * A(x).
a(n) = Sum_{k=0..floor(n/2)} (2*k)! * (k+1)^(k-1) * |Stirling1(n,2*k)|/k!.
E.g.f.: A(x) = Sum_{k>=0} (k+1)^(k-1) * log(1 - x)^(2*k) / k!.
E.g.f.: A(x) = exp( -LambertW(-log(1-x)^2) ).
E.g.f.: A(x) = -LambertW(-log(1 - x)^2)/log(1 - x)^2.
a(n) ~ sqrt(2) * exp(5/4 + exp(-1/2)/2 - n) * n^(n-1) / (1 - exp(-exp(-1/2)))^(n - 1/2). - Vaclav Kotesovec, Aug 05 2025

A357027 E.g.f. satisfies A(x) = 1/(1 - x)^(log(1 - x)^2 * A(x)).

Original entry on oeis.org

1, 0, 0, 6, 36, 210, 2430, 32424, 426552, 6575304, 118916640, 2328078456, 49421111256, 1153979875152, 29201577206256, 791744021665344, 22988121190902720, 712541051083100160, 23447653175729566080, 816434611464004145280, 30009023179153182132480
Offset: 0

Views

Author

Seiichi Manyama, Sep 09 2022

Keywords

Links

Crossrefs

Cf. A052813, A357026.

Programs

  • Mathematica
    m = 21; (* number of terms *)
A[_] = 0;
Do[A[x_] = 1/(1 - x)^(Log[1 - x]^2*A[x]) + O[x]^m // Normal, {m}];
CoefficientList[A[x], x]*Range[0, m - 1]! (* Jean-François Alcover, Sep 12 2022 *)
  • PARI
    a(n) = sum(k=0, n\3, (3*k)!*(k+1)^(k-1)*abs(stirling(n, 3*k, 1))/k!);
  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, (k+1)^(k-1)*(-log(1-x))^(3*k)/k!)))
  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(-lambertw(log(1-x)^3))))
  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(lambertw(log(1-x)^3)/log(1-x)^3))

Formula

E.g.f. satisfies log(A(x)) = -log(1 - x)^3 * A(x).
a(n) = Sum_{k=0..floor(n/3)} (3*k)! * (k+1)^(k-1) * |Stirling1(n,3*k)|/k!.
E.g.f.: A(x) = Sum_{k>=0} (k+1)^(k-1) * (-log(1 - x))^(3*k) / k!.
E.g.f.: A(x) = exp( -LambertW(log(1-x)^3) ).
E.g.f.: A(x) = LambertW(log(1 - x)^3)/log(1 - x)^3.

A375878 E.g.f. satisfies A(x) = 1/(1 - x)^(2*A(x)^(1/2)).

Original entry on oeis.org

1, 2, 10, 78, 832, 11320, 187968, 3693760, 83970640, 2170052928, 62876256000, 2019782393904, 71268840658464, 2740911076718256, 114134851494134352, 5116804468061982000, 245747690114319479808, 12589481527535031074304, 685316177026591879217664
Offset: 0

Views

Author

Seiichi Manyama, Sep 01 2024

Keywords

Links

Crossrefs

Cf. A052813, A375879.

Programs

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

Formula

E.g.f.: B(x)^2, where B(x) is the e.g.f. of A052813.
E.g.f.: exp( - 2*LambertW(log(1 - x)) ).
a(n) = 2 * Sum_{k=0..n} (k+2)^(k-1) * |Stirling1(n,k)|.
a(n) ~ 2 * n^(n-1) / ((exp(exp(-1)) - 1)^(n - 1/2) * exp(n - n*exp(-1) - 5/2)). - Vaclav Kotesovec, Aug 05 2025

A375879 E.g.f. satisfies A(x) = 1/(1 - x)^(3*A(x)^(1/3)).

Original entry on oeis.org

1, 3, 18, 159, 1860, 27180, 477702, 9830814, 232182024, 6195709008, 184478436720, 6066613989216, 218468134274904, 8553367426018896, 361834389120925224, 16450660929420051480, 800070438821317486272, 41453084674400350385664
Offset: 0

Views

Author

Seiichi Manyama, Sep 01 2024

Keywords

Links

Crossrefs

Cf. A052813, A375878.

Programs

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

Formula

E.g.f.: B(x)^3, where B(x) is the e.g.f. of A052813.
E.g.f.: exp( - 3*LambertW(log(1 - x)) ).
a(n) = 3 * Sum_{k=0..n} (k+3)^(k-1) * |Stirling1(n,k)|.
a(n) ~ 3 * n^(n-1) * exp(7/2 + n*exp(-1) - n) / (exp(exp(-1)) - 1)^(n - 1/2). - Vaclav Kotesovec, Aug 27 2025
Previous Showing 11-17 of 17 results.

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