A362572 E.g.f. satisfies A(x) = exp(x * A(x)^(x/2)).
1, 1, 1, 4, 13, 76, 421, 3361, 26209, 267688, 2689201, 33579811, 412800961, 6103089994, 88754687113, 1517513934301, 25487131948321, 495009722435176, 9430633148123809, 205154208873930763, 4371962638221712801, 105330237499426955926
Offset: 0
Keywords
Links
- Seiichi Manyama, Table of n, a(n) for n = 0..439
- Eric Weisstein's World of Mathematics, Lambert W-Function.
Programs
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PARI
my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(x*exp(-lambertw(-x^2/2)))))
Formula
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