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.

A360473 E.g.f. satisfies A(x) = exp( x * exp(x) * A(x)^2 ).

Original entry on oeis.org

1, 1, 7, 82, 1441, 34036, 1013149, 36446698, 1538703457, 74607811048, 4086635087701, 249593193648646, 16819085803158577, 1239637405609740268, 99206330021667838285, 8567230421555333516746, 794104205843228382969409
Offset: 0

Views

Author

Seiichi Manyama, Feb 08 2023

Keywords

Links

Crossrefs

Cf. A273954, A360465, A360474.

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

Formula

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

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

A360466 E.g.f. satisfies A(x) = exp(2 * x * exp(x) * A(x)).

Original entry on oeis.org

1, 2, 16, 206, 3832, 93962, 2871820, 105355406, 4515648784, 221598121490, 12257187851284, 754703476252310, 51204818674338328, 3796079000648275226, 305328667748448560668, 26483633169003911205278, 2464307301750079915255840, 244872778601760932275686434
Offset: 0

Views

Author

Seiichi Manyama, Feb 08 2023

Keywords

Links

Crossrefs

Cf. A273954, A357247, A360465.

Programs

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

Formula

E.g.f.: A(x) = exp( -LambertW(-2 * x * exp(x)) ).
E.g.f.: A(x) = -LambertW(-2 * x * exp(x)) / (2 * x * exp(x)).
E.g.f.: A(x) = Sum_{k>=0} (k+1)^(k-1) * (2 * x * exp(x))^k / k!.
a(n) = Sum_{k=0..n} 2^k * k^(n-k) * (k+1)^(k-1) * binomial(n,k).
a(n) ~ sqrt(1+LambertW(exp(-1)/2)) * n^(n-1) / (exp(n-1) * LambertW(exp(-1)/2)^n). - Vaclav Kotesovec, Feb 08 2023
Showing 1-3 of 3 results.

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