A362735 -id:A362735 - cp's OEIS frontend

A362694 E.g.f. satisfies A(x) = exp(x + x * A(x)^2).

1, 2, 12, 152, 2960, 78112, 2607808, 105432448, 5008584960, 273482293760, 16878251101184, 1161918967060480, 88277165100666880, 7337286679766179840, 662287143981044121600, 64516370031367063175168, 6746443728505612426870784, 753763691778003738319519744

Offset: 0

Seiichi Manyama, May 01 2023

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..339
Eric Weisstein's World of Mathematics, Lambert W-Function.

Cf. A349562, A362693, A362734, A362735.

Cf. A143768, A202617.

my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(x-lambertw(-2*x*exp(2*x))/2)))

E.g.f.: sqrt( -LambertW(-2*x*exp(2*x)) / (2*x) ) = exp( x - LambertW(-2*x*exp(2*x))/2 ).
a(n) = Sum_{k=0..n} (2*k+1)^(n-1) * binomial(n,k) = 2^n * A202617(n).
a(n) ~ sqrt(1 + 1/LambertW(exp(-1))) * 2^(n-1) * n^(n-1) / (exp(n) * LambertW(exp(-1))^n). - Vaclav Kotesovec, Nov 10 2023

A362734 E.g.f. satisfies A(x) = exp(x + x * A(x)^3).

Original entry on oeis.org

1, 2, 16, 296, 8512, 333632, 16595200, 1001460224, 71094759424, 5805799829504, 536188352856064, 55259197654089728, 6287146625230962688, 782751635353947865088, 105852868748672770244608, 15451195442132410179780608, 2421355190097788960505856000

Offset: 0

Seiichi Manyama, May 01 2023

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..322
Eric Weisstein's World of Mathematics, Lambert W-Function.

Cf. A349562, A362693, A362694, A362735.

Cf. A349714, A362392, A362472.

my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(x-lambertw(-3*x*exp(3*x))/3)))

E.g.f.: ( -LambertW(-3*x*exp(3*x)) / (3*x) )^(1/3) = exp( x - LambertW(-3*x*exp(3*x))/3 ).
a(n) = Sum_{k=0..n} (3*k+1)^(n-1) * binomial(n,k) = 2^n * A349714(n).
a(n) ~ sqrt(LambertW(exp(-1)) + 1) * 3^(n-1) * n^(n-1) / (exp(n) * LambertW(exp(-1))^(n + 1/3)). - Vaclav Kotesovec, Apr 24 2024

A362693 E.g.f. satisfies A(x) = exp(x + x / A(x)).

Original entry on oeis.org

1, 2, 0, 8, -64, 832, -13568, 269824, -6328320, 171044864, -5235245056, 178988498944, -6760886435840, 279614956503040, -12566949343002624, 609881495812702208, -31785828867471572992, 1770660964785178279936, -104990165030126886060032

Offset: 0

Seiichi Manyama, May 01 2023

sign

Seiichi Manyama, Table of n, a(n) for n = 0..372
Eric Weisstein's World of Mathematics, Lambert W-Function.

Cf. A349562, A362694, A362734, A362735.
Cf. A362736, A362737.
Cf. A349719.

nmax = 20; A[_] = 1;
Do[A[x_] = Exp[x + x/A[x]] + O[x]^(nmax+1) // Normal, {nmax}];
CoefficientList[A[x], x]*Range[0, nmax]! (* Jean-François Alcover, Mar 04 2024 *)

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

E.g.f.: x / LambertW(x*exp(-x)) = exp( x + LambertW(x*exp(-x)) ).

a(n) = Sum_{k=0..n} (-k+1)^(n-1) * binomial(n,k) = 2^n * A349719(n).

cp's OEIS Frontend

A362694 E.g.f. satisfies A(x) = exp(x + x * A(x)^2).

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Formula

A362734 E.g.f. satisfies A(x) = exp(x + x * A(x)^3).

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Formula

A362693 E.g.f. satisfies A(x) = exp(x + x / A(x)).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula