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 1-4 of 4 results.

A362474 E.g.f. satisfies A(x) = exp(x + x^2/2 * A(x)^2).

Original entry on oeis.org

1, 1, 2, 10, 70, 646, 7576, 106744, 1761628, 33361948, 712950616, 16976294776, 445751093800, 12795850109992, 398697898011232, 13401365473319776, 483376669737381136, 18623161719254837008, 763300232417720682784, 33163224556779213475744
Offset: 0

Views

Author

Seiichi Manyama, Apr 21 2023

Keywords

Links

Crossrefs

Column k=1 of A362483.
Cf. A362478, A362491.
Cf. A143768, A362475.

Programs

  • Mathematica
    nmax = 20; A[_] = 1;
Do[A[x_] = Exp[x + x^2/2*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(x-lambertw(-x^2*exp(2*x))/2)))

Formula

E.g.f.: exp(x - LambertW(-x^2 * exp(2*x))/2) = sqrt(-LambertW(-x^2*exp(2*x))/x^2).
a(n) = n! * Sum_{k=0..floor(n/2)} (1/2)^k * (2*k+1)^(n-k-1) / (k! * (n-2*k)!).
a(n) ~ sqrt(1 + LambertW(exp(-1/2))) * n^(n-1) / (sqrt(2) * exp(n) * LambertW(exp(-1/2))^(n+1)). - Vaclav Kotesovec, Nov 10 2023

A362478 E.g.f. satisfies A(x) = exp(x + x^3/3 * A(x)^3).

Original entry on oeis.org

1, 1, 1, 3, 33, 321, 2841, 31641, 498849, 8979489, 167510961, 3427780401, 80374833441, 2089382321313, 58020408889353, 1721768971537161, 55150870311938241, 1897482353016075201, 69322763655015214689, 2676706914491568918369
Offset: 0

Views

Author

Seiichi Manyama, Apr 21 2023

Keywords

Links

Crossrefs

Column k=2 of A362490.
Cf. A362474, A362491.
Cf. A362390.

Programs

  • Mathematica
    nmax = 20; A[_] = 1;
Do[A[x_] = Exp[x + x^3/3*A[x]^3] + 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(x-lambertw(-x^3*exp(3*x))/3)))

Formula

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

A362473 E.g.f. satisfies A(x) = exp(x + x^4 * A(x)^4).

Original entry on oeis.org

1, 1, 1, 1, 25, 601, 9001, 105001, 1231441, 24146641, 740098801, 22443260401, 607394284201, 16102368745321, 497289446373721, 19072987370400601, 806135144596672801, 33945128330918599201, 1426006261391514829921, 63478993000497055809121
Offset: 0

Views

Author

Seiichi Manyama, Apr 21 2023

Keywords

Links

Crossrefs

Cf. A143768, A349562, A362472.
Cf. A362393, A362482, A362491.

Programs

  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(x-lambertw(-4*x^4*exp(4*x))/4)))

Formula

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

A362494 E.g.f. satisfies A(x) = exp(x - x^4/4 * A(x)^4).

Original entry on oeis.org

1, 1, 1, 1, -5, -149, -2249, -26249, -251159, -1443959, 21646801, 1209344401, 35457894451, 817789456771, 14796993881671, 137893562065351, -4661597156689199, -372730180154530799, -16419790692323174879, -559989133713039523679, -14492546886670841884949
Offset: 0

Views

Author

Seiichi Manyama, Apr 22 2023

Keywords

Links

Crossrefs

Cf. A362492, A362493.
Cf. A362491.

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(x-lambertw(x^4*exp(4*x))/4)))

Formula

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

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