A355789 - OEIS

%I #11 Jul 18 2022 12:32:42

%S 1,1,1,2,9,52,363,3082,30817,353640,4582451,66201126,1055059569,

%T 18388749628,347959910171,7104264359810,155670829426113,

%U 3644019928871376,90755590315468003,2396199304577668190,66855611152288637713,1965490144910199279780

%N E.g.f. satisfies log(A(x)) = (1 - exp(-2*x)) * A(x)/2.

%F E.g.f.: exp( -LambertW((exp(-2*x) - 1)/2) ).

%F a(n) = Sum_{k=0..n} (-2)^(n-k) * (k+1)^(k-1) * Stirling2(n,k).

%F From _Vaclav Kotesovec_, Jul 18 2022: (Start)

%F E.g.f.: 2*LambertW((exp(-2*x) - 1)/2) / (exp(-2*x) - 1).

%F 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)

%o (PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(-lambertw((exp(-2*x)-1)/2))))

%o (PARI) a(n) = sum(k=0, n, (-2)^(n-k)*(k+1)^(k-1)*stirling(n, k, 2));

%Y Cf. A058864, A355782.

%K nonn

%O 0,4

%A _Seiichi Manyama_, Jul 17 2022