A355789 E.g.f. satisfies log(A(x)) = (1 - exp(-2*x)) * A(x)/2.
1, 1, 1, 2, 9, 52, 363, 3082, 30817, 353640, 4582451, 66201126, 1055059569, 18388749628, 347959910171, 7104264359810, 155670829426113, 3644019928871376, 90755590315468003, 2396199304577668190, 66855611152288637713, 1965490144910199279780
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Keywords
Programs
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PARI
my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(-lambertw((exp(-2*x)-1)/2)))) -
PARI
a(n) = sum(k=0, n, (-2)^(n-k)*(k+1)^(k-1)*stirling(n, k, 2));
Formula
E.g.f.: exp( -LambertW((exp(-2*x) - 1)/2) ).
a(n) = Sum_{k=0..n} (-2)^(n-k) * (k+1)^(k-1) * Stirling2(n,k).
From Vaclav Kotesovec, Jul 18 2022: (Start)
E.g.f.: 2*LambertW((exp(-2*x) - 1)/2) / (exp(-2*x) - 1).
a(n) ~ sqrt(exp(1)/2 - 1) * sqrt(-log(1 - 2*exp(-1))) * 2^n * n^(n-1) / (exp(n-1) * (1 - log(exp(1) - 2))^n). (End)