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.

A360547 E.g.f. satisfies A(x) = exp( x * ( exp(x) * A(x) )^2 ).

Original entry on oeis.org

1, 1, 9, 121, 2417, 64721, 2180665, 88719625, 4233968737, 231991022881, 14356691152361, 990506937621785, 75390334060230865, 6275675303410022641, 567191776288882702105, 55313848534122299876521, 5789703106014903009828545, 647414950001156861671249985
Offset: 0

Views

Author

Seiichi Manyama, Feb 11 2023

Keywords

Links

Crossrefs

Cf. A273953, A273954, A360544.
Cf. A360465, A360473.

Programs

  • Mathematica
    nmax = 20; A[_] = 1;
Do[A[x_] = Exp[x*(Exp[x]*A[x])^2] + 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(exp(-lambertw(-2*x*exp(2*x))/2)))
  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(sqrt(-lambertw(-2*x*exp(2*x))/(2*x*exp(2*x)))))
  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(sqrt(sum(k=0, N, (k+1)^(k-1)*(2*x*exp(2*x))^k/k!))))
  • PARI
    a(n) = sum(k=0, n, (2*k)^(n-k)*(2*k+1)^(k-1)*binomial(n, k));

Formula

E.g.f.: A(x) = exp( (-1/2) * LambertW(-2*x * exp(2*x)) ).
E.g.f.: A(x) = sqrt( -LambertW(-2*x * exp(2*x)) / (2*x * exp(2*x)) ).
E.g.f.: A(x) = sqrt( Sum_{k>=0} (k+1)^(k-1) * (2*x * exp(2*x))^k / k! ).
a(n) = Sum_{k=0..n} (2*k)^(n-k) * (2*k+1)^(k-1) * binomial(n,k).
a(n) ~ sqrt(1 + LambertW(exp(-1))) * 2^(n-1) * n^(n-1) / (exp(n - 1/2) * LambertW(exp(-1))^n). - Vaclav Kotesovec, Feb 17 2023

A360545 E.g.f. satisfies A(x) = x * exp( 3*(x + A(x))/2 ).

Original entry on oeis.org

0, 1, 6, 54, 756, 14580, 358668, 10736712, 378823392, 15395255280, 708217959600, 36380741745744, 2064234271203360, 128214974795177088, 8652900673357097472, 630483717450225530880, 49330027417316557012992, 4124992361928178722764544
Offset: 0

Views

Author

Seiichi Manyama, Feb 11 2023

Keywords

Links

Crossrefs

Cf. A100526, A216857, A360548.
Cf. A360482, A360544.

Programs

  • PARI
    my(N=20, x='x+O('x^N)); concat(0, Vec(serlaplace(-2*lambertw(-3*x/2*exp(3*x/2))/3)))
  • PARI
    a(n) = sum(k=1, n, (3*k/2)^(n-1)*binomial(n, k));

Formula

E.g.f.: A(x) = (-2/3) * LambertW(-3*x/2 * exp(3*x/2)).
a(n) = Sum_{k=1..n} (3*k/2)^(n-1) * binomial(n,k) = 3^(n-1) * A100526(n).
a(n) ~ sqrt(1 + LambertW(exp(-1))) * 3^(n-1) * n^(n-1) / (2^(n-1) * exp(n) * LambertW(exp(-1))^n). - Vaclav Kotesovec, Feb 17 2023

A372279 E.g.f. A(x) satisfies A(x) = exp( x * ( exp(x) * A(x) )^(5/2) ).

Original entry on oeis.org

1, 1, 11, 181, 4461, 148101, 6202651, 314158461, 18682884681, 1276509416761, 98552772971451, 8485633118339301, 806247602665104661, 83796784405535693181, 9457590223483413296811, 1151924494605809502276301, 150602291336042725831941201
Offset: 0

Views

Author

Seiichi Manyama, Apr 25 2024

Keywords

Links

Crossrefs

Cf. A273953, A273954, A360544, A360547.
Cf. A372278.

Programs

  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(-2/5*lambertw(-5/2*x*exp(5*x/2)))))
  • PARI
    a(n) = sum(k=0, n, (5*k/2)^(n-k)*(5*k/2+1)^(k-1)*binomial(n, k));

Formula

E.g.f.: A(x) = exp( -2/5 * LambertW(-5*x/2 * exp(5*x/2)) ).
E.g.f.: A(x) = ( -LambertW(-5*x/2 * exp(5*x/2)) / (5*x/2 * exp(5*x/2)) )^(2/5).
E.g.f.: A(x) = ( Sum_{k>=0} (k+1)^(k-1) * (5*x/2 * exp(5*x/2))^k / k! )^(2/5).
a(n) = Sum_{k=0..n} (5*k/2)^(n-k) * (5*k/2+1)^(k-1) * binomial(n,k).
a(n) ~ sqrt(1 + LambertW(exp(-1))) * 5^(n-1) * n^(n-1) / (exp(n - 2/5) * 2^(n-1) * LambertW(exp(-1))^n). - Vaclav Kotesovec, May 06 2024
Showing 1-3 of 3 results.

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

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