A355782 - OEIS

%I #16 Jul 18 2022 12:01:22

%S 1,2,10,94,1314,24494,572418,16109678,530772610,20049256686,

%T 854425665410,40560727143534,2122785621956226,121440903560075246,

%U 7539867236251002242,504946360197545803630,36284349255747713008770,2784785703026225861819118

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

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

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

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

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

%F a(n) ~ sqrt(2*exp(1) - 1) * sqrt(log(2/(2 - exp(-1)))) * n^(n-1) / (exp(n-1) * (log(2/(2*exp(1)-1)) + 1)^n). (End)

%o (PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(-lambertw(2*(exp(-x)-1)))))

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

%Y Cf. A058864, A351277, A355789.

%K nonn

%O 0,2

%A _Seiichi Manyama_, Jul 16 2022