A356927 - OEIS

%I #19 Nov 14 2022 10:13:03

%S 1,1,6,54,676,10980,220488,5289592,147828896,4721152320,169723566240,

%T 6785559484704,298726260001728,14362141350822720,748845960914596608,

%U 42092072779399215360,2537464961261745635328,163317885950059143238656

%N E.g.f. satisfies A(x) = 1/(1 - x)^(A(x)/(1 - x)).

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/LambertW-Function.html">Lambert W-Function</a>.

%F E.g.f.: A(x) = Sum_{k>=0} (k+1)^(k-1) * (-log(1-x)/(1-x))^k / k!.

%F E.g.f.: A(x) = exp( -LambertW(log(1-x)/(1-x)) ).

%F E.g.f.: A(x) = (1-x) * LambertW(log(1-x)/(1-x))/log(1-x).

%F a(n) ~ sqrt(1 + LambertW(exp(-1))) * n^(n-1) / (LambertW(exp(-1)) * exp(n - 1/2) * (1 - exp(1)*LambertW(exp(-1)))^(n - 1/2)). - _Vaclav Kotesovec_, Nov 14 2022

%o (PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, (k+1)^(k-1)*(-log(1-x)/(1-x))^k/k!)))

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

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

%Y Cf. A000254, A000272, A052813, A087761, A356925.

%K nonn

%O 0,3

%A _Seiichi Manyama_, Sep 04 2022