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.

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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)).

A356949 E.g.f. satisfies log(A(x)) = x^2 * (exp(x) - 1) * A(x).

Original entry on oeis.org

1, 0, 0, 6, 12, 20, 1110, 7602, 35336, 1103832, 14984010, 134552990, 3457329612, 70828191876, 1017237973934, 25648737955050, 676111332667920, 13760810592066992, 373071111301807506, 11594147432172228918, 307097278689726728660, 9330499711181779575900
Offset: 0

Views

Author

Seiichi Manyama, Sep 06 2022

Keywords

Links

Crossrefs

Cf. A052880, A355843, A356950.
Cf. A000272, A240989, A355287, A356797, A356951.

Programs

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

Formula

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

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