A362569 -id:A362569 - 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!.

A362568 E.g.f. satisfies A(x) = exp(x/A(x)^x).

Original entry on oeis.org

1, 1, 1, -5, -23, 121, 1321, -7349, -148175, 853777, 27840241, -163354949, -7934320679, 46820981065, 3203091569497, -18833438286389, -1742847946697759, 10137524365568161, 1230956201929018465, -7042544858204663813, -1095864481054115534519

Offset: 0

Seiichi Manyama, Apr 25 2023

sign

Seiichi Manyama, Table of n, a(n) for n = 0..417
Eric Weisstein's World of Mathematics, Lambert W-Function.

Cf. A177885, A362569.

Cf. A361777.

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

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

cp's OEIS Frontend

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

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