A362691 -id:A362691 - cp's OEIS frontend

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

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)!).

A362748 E.g.f. satisfies A(x) = exp(x^3/6 + x * A(x)).

Original entry on oeis.org

1, 1, 3, 17, 133, 1386, 18097, 284299, 5225985, 110097836, 2616190831, 69236871309, 2019833025157, 64403044165942, 2228441614038837, 83166830262851591, 3330183199746011713, 142418071427679810936, 6478769455582913796475

Offset: 0

Seiichi Manyama, May 02 2023

nonn

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

Cf. A349562, A362747.

Cf. A362381, A362691.

my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(x^3/6-lambertw(-x*exp(x^3/6)))))

E.g.f.: -LambertW(-x * exp(x^3/6)) / x = exp( x^3/6 - LambertW(-x*exp(x^3/6)) ).

a(n) = n! * Sum_{k=0..floor(n/3)} (n-3*k+1)^(n-2*k-1) / (6^k * k! * (n-3*k)!).

cp's OEIS Frontend

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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A362748 E.g.f. satisfies A(x) = exp(x^3/6 + x * A(x)).

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