A362573 -id:A362573 - cp's OEIS frontend

A362571 E.g.f. satisfies A(x) = exp(x * A(x)^(x^2)).

1, 1, 1, 1, 25, 121, 361, 8401, 82321, 456625, 11496241, 172149121, 1452983401, 40947003241, 823437038425, 9491714865361, 300842942443681, 7568303382376801, 111494036396244961, 3957438528527140225, 119206427681076135481, 2147109997071581380441

Offset: 0

Seiichi Manyama, Apr 25 2023

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..427
Eric Weisstein's World of Mathematics, Lambert W-Function.

Cf. A000272, A361777.

Cf. A362569, A362573.

my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(x*exp(-lambertw(-x^3)))))

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

A362572 E.g.f. satisfies A(x) = exp(x * A(x)^(x/2)).

Original entry on oeis.org

1, 1, 1, 4, 13, 76, 421, 3361, 26209, 267688, 2689201, 33579811, 412800961, 6103089994, 88754687113, 1517513934301, 25487131948321, 495009722435176, 9430633148123809, 205154208873930763, 4371962638221712801, 105330237499426955926

Offset: 0

Seiichi Manyama, Apr 25 2023

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..439
Eric Weisstein's World of Mathematics, Lambert W-Function.

Cf. A000272, A362573.

Cf. A361777.

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

E.g.f.: (-2 * LambertW(-x^2/2) / x^2)^(2/x) = exp(-2 * LambertW(-x^2/2) / x) = exp(x * exp(-LambertW(-x^2/2))).
a(n) = n! * Sum_{k=0..floor(n/2)} ((n-k)/2)^k * binomial(n-k-1,k)/(n-k)!.
E.g.f.: Sum_{k>=0} (k*x/2 + 1)^(k-1) * x^k / k!.
a(n) ~ (exp(sqrt(2)*exp(1/2)) - (-1)^n*exp(-sqrt(2)*exp(1/2))) * n^(n-1) / (2^(n/2 - 1) * exp((n-1)/2)). - Vaclav Kotesovec, Aug 05 2025

cp's OEIS Frontend

A362571 E.g.f. satisfies A(x) = exp(x * A(x)^(x^2)).

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