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 1-5 of 5 results.

A356910 E.g.f. satisfies A(x)^A(x) = 1/(1 - x)^(x^2).

Original entry on oeis.org

1, 0, 0, 6, 12, 40, -180, -1512, -11760, 142560, 2701440, 37033920, -47472480, -7299227520, -181704466944, -904179830400, 40024286265600, 1774386897454080, 24426730612869120, -217650777809310720, -26326923875473536000, -662608157128469637120
Offset: 0

Views

Author

Seiichi Manyama, Sep 03 2022

Keywords

Links

Crossrefs

Cf. A356905, A356911.
Cf. A353228, A355287.

Programs

  • Mathematica
    nmax = 21; A[_] = 1;
Do[A[x_] = ((1 - x)^(-x^2))^(1/A[x]) + O[x]^(nmax+1) // Normal, {nmax}];
CoefficientList[A[x], x]*Range[0, nmax]! (* Jean-François Alcover, Mar 04 2024 *)
  • PARI
    a(n) = n!*sum(k=0, n\3, (-k+1)^(k-1)*abs(stirling(n-2*k, k, 1))/(n-2*k)!);
  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, (-k+1)^(k-1)*(-x^2*log(1-x))^k/k!)))
  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(lambertw(-x^2*log(1-x)))))
  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(-x^2*log(1-x)/lambertw(-x^2*log(1-x))))

Formula

a(n) = n! * Sum_{k=0..floor(n/3)} (-k+1)^(k-1) * |Stirling1(n-2*k,k)|/(n-2*k)!.
E.g.f.: A(x) = Sum_{k>=0} (-k+1)^(k-1) * (-x^2 * log(1-x))^k / k!.
E.g.f.: A(x) = exp( LambertW(-x^2 * log(1-x)) ).
E.g.f.: A(x) = -x^2 * log(1-x)/LambertW(-x^2 * log(1-x)).

A356911 E.g.f. satisfies A(x)^A(x) = 1/(1 - x)^(x^3).

Original entry on oeis.org

1, 0, 0, 0, 24, 60, 240, 1260, -12096, -120960, -1144800, -11642400, 190270080, 4670265600, 81378198720, 1348668921600, -880532674560, -406217626214400, -13255586359142400, -343166884178227200, -3137937973466572800, 72862796986940620800
Offset: 0

Views

Author

Seiichi Manyama, Sep 03 2022

Keywords

Links

Crossrefs

Cf. A356905, A356910.
Cf. A353229, A356099.

Programs

  • Mathematica
    nmax = 21; A[_] = 1;
Do[A[x_] = ((1 - x)^(-x^3))^(1/A[x]) + O[x]^(nmax+1) // Normal, {nmax}];
CoefficientList[A[x], x]*Range[0, nmax]! (* Jean-François Alcover, Mar 04 2024 *)
  • PARI
    a(n) = n!*sum(k=0, n\4, (-k+1)^(k-1)*abs(stirling(n-3*k, k, 1))/(n-3*k)!);
  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, (-k+1)^(k-1)*(-x^3*log(1-x))^k/k!)))
  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(lambertw(-x^3*log(1-x)))))
  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(-x^3*log(1-x)/lambertw(-x^3*log(1-x))))

Formula

a(n) = n! * Sum_{k=0..floor(n/4)} (-k+1)^(k-1) * |Stirling1(n-3*k,k)|/(n-3*k)!.
E.g.f.: A(x) = Sum_{k>=0} (-k+1)^(k-1) * (-x^3 * log(1-x))^k / k!.
E.g.f.: A(x) = exp( LambertW(-x^3 * log(1-x)) ).
E.g.f.: A(x) = -x^3 * log(1-x)/LambertW(-x^3 * log(1-x)).

A356913 E.g.f. satisfies A(x)^A(x) = 1/(1 - x)^(x^3/6).

Original entry on oeis.org

1, 0, 0, 0, 4, 10, 40, 210, 784, 5040, 40200, 369600, 5285280, 72072000, 1006889520, 14760345600, 210510263040, 3131345817600, 49229619129600, 818940523564800, 15054020163619200, 301204611031564800, 6455999452413772800, 146587705490513548800
Offset: 0

Views

Author

Seiichi Manyama, Sep 03 2022

Keywords

Links

Crossrefs

Cf. A356905, A356912.
Cf. A351493, A356753.

Programs

  • Mathematica
    nmax = 23; A[_] = 1;
Do[A[x_] = ((1 - x)^(-x^3/6))^(1/A[x]) + O[x]^(nmax+1) // Normal, {nmax}];
CoefficientList[A[x], x]*Range[0, nmax]! (* Jean-François Alcover, Mar 04 2024 *)
  • PARI
    a(n) = n!*sum(k=0, n\4, (-k+1)^(k-1)*abs(stirling(n-3*k, k, 1))/(6^k*(n-3*k)!));
  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, (-k+1)^(k-1)*(-x^3/6*log(1-x))^k/k!)))
  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(lambertw(-x^3/6*log(1-x)))))
  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(-x^3/6*log(1-x)/lambertw(-x^3/6*log(1-x))))

Formula

a(n) = n! * Sum_{k=0..floor(n/4)} (-k+1)^(k-1) * |Stirling1(n-3*k,k)|/(6^k * (n-3*k)!).
E.g.f.: A(x) = Sum_{k>=0} (-k+1)^(k-1) * (-x^3/6 * log(1-x))^k / k!.
E.g.f.: A(x) = exp( LambertW(-x^3/6 * log(1-x)) ).
E.g.f.: A(x) = -x^3/6 * log(1-x)/LambertW(-x^3/6 * log(1-x)).

A357243 E.g.f. satisfies A(x)^A(x) = 1/(1 - x)^(1 - x).

Original entry on oeis.org

1, 1, -2, 6, -52, 540, -7608, 129304, -2612608, 60867360, -1608663840, 47527158624, -1552431588288, 55547889458880, -2160724031160576, 90782738645280000, -4097139872604807168, 197675862365363088384, -10153243488783257091072
Offset: 0

Views

Author

Seiichi Manyama, Sep 19 2022

Keywords

Links

Crossrefs

Cf. A005727, A155456, A349561, A356905, A356908.

Programs

  • Mathematica
    nmax = 20; A[_] = 1;
Do[A[x_] = ((1 - x)^(-1 + x))^(1/A[x]) + O[x]^(nmax+1) // Normal, {nmax}];
CoefficientList[A[x], x]*Range[0, nmax]! (* Jean-François Alcover, Mar 04 2024 *)
  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, (-k+1)^(k-1)*(-(1-x)*log(1-x))^k/k!)))
  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(lambertw(-(1-x)*log(1-x)))))
  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(-(1-x)*log(1-x)/lambertw(-(1-x)*log(1-x))))

Formula

E.g.f. satisfies A(x)^A(x) * (1 - x)^(1 - x) = 1.
E.g.f.: A(x) = Sum_{k>=0} (-k+1)^(k-1) * (-(1-x) * log(1-x))^k / k!.
E.g.f.: A(x) = exp( LambertW(-(1-x) * log(1-x)) ).
E.g.f.: A(x) = -(1-x) * log(1-x)/LambertW(-(1-x) * log(1-x)).

A356906 E.g.f. satisfies A(x)^(A(x)^2) = 1/(1 - x)^x.

Original entry on oeis.org

1, 0, 2, 3, -28, -150, 2154, 26040, -322512, -7872984, 77570280, 3752301960, -22068935736, -2542757920560, 1422846762960, 2302464947491800, 14860063644794880, -2653728770258072640, -41790782141846648640, 3739260018343338345600
Offset: 0

Views

Author

Seiichi Manyama, Sep 03 2022

Keywords

Crossrefs

Cf. A355842, A356795, A356796, A356905.
Cf. A349652, A356885.

Programs

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

Formula

a(n) = n! * Sum_{k=0..floor(n/2)} (-2*k+1)^(k-1) * |Stirling1(n-k,k)|/(n-k)!.
E.g.f.: A(x) = Sum_{k>=0} (-2*k+1)^(k-1) * (-x * log(1-x))^k / k!.
E.g.f.: A(x) = exp( LambertW(-2 * x * log(1-x))/2 ).
E.g.f.: A(x) = ( -2 * x * log(1-x)/LambertW(-2 * x * log(1-x)) )^(1/2).
Showing 1-5 of 5 results.

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