A362737 -id:A362737 - cp's OEIS frontend

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

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).

A362691 E.g.f. satisfies A(x) = exp(x^3 + x * A(x)).

Original entry on oeis.org

1, 1, 3, 22, 173, 1836, 24847, 403474, 7667865, 167097016, 4108985531, 112562882334, 3399748630357, 112246652293972, 4022094151907847, 155461592488721866, 6447531477912609713, 285606134199075271536, 13458367778796518816755

Offset: 0

Seiichi Manyama, May 01 2023

nonn

Eric Weisstein's World of Mathematics, Lambert W-Function.

Cf. A349562, A362690.

Cf. A362737.

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

E.g.f.: -LambertW(-x * exp(x^3)) / x = exp( x^3 - LambertW(-x*exp(x^3)) ).

a(n) = n! * Sum_{k=0..floor(n/3)} (n-3*k+1)^(n-2*k-1) / (k! * (n-3*k)!).

A362736 E.g.f. satisfies A(x) = exp(x^2 + x / A(x)).

Original entry on oeis.org

1, 1, 1, 4, -3, 96, -755, 10368, -147623, 2492416, -47137959, 996741120, -23260103339, 594198429696, -16492683271259, 494278721929216, -15908038836914895, 547238863907586048, -20038031401448021327, 778147549666716155904, -31943565308583934360019

Offset: 0

Seiichi Manyama, May 01 2023

sign

Eric Weisstein's World of Mathematics, Lambert W-Function.

Cf. A362693, A362737.

Cf. A362690.

my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(x^2+lambertw(x*exp(-x^2)))))

E.g.f.: x / LambertW(x*exp(-x^2)) = exp( x^2 + LambertW(x*exp(-x^2)) ).

a(n) = n! * Sum_{k=0..floor(n/2)} (-n+2*k+1)^(n-k-1) / (k! * (n-2*k)!).

cp's OEIS Frontend

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

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A362691 E.g.f. satisfies A(x) = exp(x^3 + x * A(x)).

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A362736 E.g.f. satisfies A(x) = exp(x^2 + x / A(x)).

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