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 1-3 of 3 results.

A356951 E.g.f. satisfies log(A(x)) = x^2/2 * (exp(x) - 1) * A(x).

Original entry on oeis.org

1, 0, 0, 3, 6, 10, 285, 1911, 8848, 155016, 1931625, 17006275, 276807036, 4801114968, 65672925409, 1172625764415, 24657199159440, 460156401399376, 9560083801337793, 230955040794126915, 5393971086379904260, 131545127670380245920, 3587507216606547324321
Offset: 0

Views

Author

Seiichi Manyama, Sep 06 2022

Keywords

Links

Crossrefs

Cf. A052880, A355843, A356952.
Cf. A000272, A354000, A356752, A356949.

Programs

  • Mathematica
    nmax = 22; A[_] = 1;
Do[A[x_] = Exp[x^2/2*(Exp[x] - 1)*A[x]] + O[x]^(nmax+1) // Normal, {nmax}];
CoefficientList[A[x], x]*Range[0, nmax]! (* Jean-François Alcover, Mar 05 2024 *)
  • PARI
    a(n) = n!*sum(k=0, n\3, (k+1)^(k-1)*stirling(n-2*k, k, 2)/(2^k*(n-2*k)!));
  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, (k+1)^(k-1)*(x^2/2*(exp(x)-1))^k/k!)))
  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(-lambertw(x^2/2*(1-exp(x))))))
  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(lambertw(x^2/2*(1-exp(x)))/(x^2/2*(1-exp(x)))))

Formula

a(n) = n! * Sum_{k=0..floor(n/3)} (k+1)^(k-1) * Stirling2(n-2*k,k)/(2^k * (n-2*k)!).
E.g.f.: A(x) = Sum_{k>=0} (k+1)^(k-1) * (x^2/2 * (exp(x) - 1))^k / k!.
E.g.f.: A(x) = exp( -LambertW(x^2/2 * (1 - exp(x))) ).
E.g.f.: A(x) = LambertW(x^2/2 * (1 - exp(x)))/(x^2/2 * (1 - exp(x))).

A356963 E.g.f. satisfies log(A(x)) = x^3/6 * (exp(x*A(x)) - 1) * A(x).

Original entry on oeis.org

1, 0, 0, 0, 4, 10, 20, 35, 2856, 30324, 205920, 1108965, 23266540, 482882686, 6894708184, 75981906455, 1191424314640, 28555511625320, 635370500529216, 11547272842560969, 213976417787571540, 5165550420447656050, 142251473243071866520, 3624660458114853235771
Offset: 0

Views

Author

Seiichi Manyama, Sep 06 2022

Keywords

Crossrefs

Cf. A349557, A356785, A356962.
Cf. A356952, A356968.

Programs

  • PARI
    a(n) = n!*sum(k=0, n\4, (n-2*k+1)^(k-1)*stirling(n-3*k, k, 2)/(6^k*(n-3*k)!));

Formula

a(n) = n! * Sum_{k=0..floor(n/4)} (n-2*k+1)^(k-1) * Stirling2(n-3*k,k)/(6^k * (n-3*k)!).

A356950 E.g.f. satisfies log(A(x)) = x^3 * (exp(x) - 1) * A(x).

Original entry on oeis.org

1, 0, 0, 0, 24, 60, 120, 210, 60816, 544824, 3175920, 14969790, 1339209960, 25141598196, 291418089144, 2618105492730, 128974591028640, 3841451570440560, 73103023032142176, 1060951475511351414, 39132892925113341240, 1516348247446904304300
Offset: 0

Views

Author

Seiichi Manyama, Sep 06 2022

Keywords

Links

Crossrefs

Cf. A052880, A355843, A356949.
Cf. A000272, A356099, A356952.

Programs

  • Mathematica
    nmax = 21; A[_] = 1;
Do[A[x_] = Exp[(-1 + Exp[x])*A[x]*x^3] + 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)*stirling(n-3*k, k, 2)/(n-3*k)!);
  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, (k+1)^(k-1)*(x^3*(exp(x)-1))^k/k!)))
  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(-lambertw(x^3*(1-exp(x))))))
  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(lambertw(x^3*(1-exp(x)))/(x^3*(1-exp(x)))))

Formula

a(n) = n! * Sum_{k=0..floor(n/4)} (k+1)^(k-1) * Stirling2(n-3*k,k)/(n-3*k)!.
E.g.f.: A(x) = Sum_{k>=0} (k+1)^(k-1) * (x^3 * (exp(x) - 1))^k / k!.
E.g.f.: A(x) = exp( -LambertW(x^3 * (1 - exp(x))) ).
E.g.f.: A(x) = LambertW(x^3 * (1 - exp(x)))/(x^3 * (1 - exp(x))).
Showing 1-3 of 3 results.

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