A376104 Expansion of e.g.f. -LambertW(-2*x / (1 + x))/2.
0, 1, 2, 18, 200, 3240, 65712, 1626352, 47357312, 1587917952, 60244640000, 2551693841664, 119354176490496, 6110496488651776, 339867366232131584, 20407634663085066240, 1315738882989816578048, 90655729379062051799040, 6647791273573299221495808
Offset: 0
Keywords
Links
- Eric Weisstein's World of Mathematics, Lambert W-Function.
Programs
-
PARI
my(N=20, x='x+O('x^N)); concat(0, Vec(serlaplace(-lambertw(-2*x/(1+x))/2))) -
PARI
a(n) = n!*sum(k=1, n, (-1)^(n-k)*(2*k)^(k-1)*binomial(n-1, k-1)/k!);
Formula
E.g.f. A(x) satisfies A(x) = x * (-A(x) + exp(2*A(x))).
E.g.f.: Series_Reversion( x / (-x + exp(2*x)) ).
a(n) = n! * Sum_{k=1..n} (-1)^(n-k) * (2*k)^(k-1) * binomial(n-1,k-1)/k!.
a(n) ~ (2-exp(-1))^(n + 1/2) * n^(n-1) / 2^(3/2). - Vaclav Kotesovec, Sep 11 2024