login
Expansion of e.g.f. -LambertW(-x / (1 - 2*x)).
3

%I #13 Sep 10 2024 08:04:25

%S 0,1,6,57,760,13265,289116,7600873,234730224,8340307137,335388171700,

%T 15062758093361,747393408423432,40606032733746961,2397539426985311532,

%U 152864047998089113785,10467226142002168282336,766094017043351707135745

%N Expansion of e.g.f. -LambertW(-x / (1 - 2*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) satisfies A(x) = x * (2*A(x) + exp(A(x))).

%F E.g.f.: Series_Reversion( x / (2*x + exp(x)) ).

%F a(n) = n! * Sum_{k=1..n} 2^(n-k) * k^(k-1) * binomial(n-1,k-1)/k!.

%F a(n) ~ (1 + 2*exp(-1))^(n + 1/2) * n^(n-1). - _Vaclav Kotesovec_, Sep 10 2024

%o (PARI) my(N=20, x='x+O('x^N)); concat(0, Vec(serlaplace(-lambertw(-x/(1-2*x)))))

%o (PARI) a(n) = n!*sum(k=1, n, 2^(n-k)*k^(k-1)*binomial(n-1, k-1)/k!);

%Y Cf. A052871, A376101.

%K nonn

%O 0,3

%A _Seiichi Manyama_, Sep 10 2024