A362736 -id:A362736 - 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).

A362690 E.g.f. satisfies A(x) = exp(x^2 + x * A(x)).

Original entry on oeis.org

1, 1, 5, 28, 245, 2816, 40537, 702976, 14270153, 332102656, 8719631981, 255020847104, 8222803663549, 289815184113664, 11085650268060929, 457386463819595776, 20248713707077863953, 957435459515190345728, 48157934732749633188565

Offset: 0

Seiichi Manyama, May 01 2023

nonn

Essentially the same as A138293.

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

Cf. A349562, A362691.

Cf. A125500, A138293, A362736.

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

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

A362737 E.g.f. satisfies A(x) = exp(x^3 + x / A(x)).

Original entry on oeis.org

1, 1, -1, 10, -27, 316, -3725, 63666, -1177687, 25196536, -607345209, 16391726110, -488872392371, 15968546353332, -566886190710853, 21733419523383946, -894910999976666415, 39390009619800983536, -1845602126785662907121, 91714859182521808208694

Offset: 0

Seiichi Manyama, May 01 2023

sign

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

Cf. A362693, A362736.

Cf. A362691.

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

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

cp's OEIS Frontend

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

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A362690 E.g.f. satisfies A(x) = exp(x^2 + x * A(x)).

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A362737 E.g.f. satisfies A(x) = exp(x^3 + x / A(x)).

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