A376099 Expansion of e.g.f. -LambertW(-3*x / (1 - x))/3.
0, 1, 8, 123, 2940, 96465, 4035438, 205395687, 12320780328, 851216818977, 66565617543450, 5812559883272439, 560602050420898764, 59186681025383491281, 6789351417468526481526, 840843424588323640992615, 111820607202879512913388752, 15892724010727366554445999425
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(-3*x/(1-x))/3))) -
PARI
a(n) = n!*sum(k=1, n, (3*k)^(k-1)*binomial(n-1, k-1)/k!);
Formula
E.g.f. A(x) satisfies A(x) = x * (A(x) + exp(3*A(x))).
E.g.f.: Series_Reversion( x / (x + exp(3*x)) ).
a(n) = n! * Sum_{k=1..n} (3*k)^(k-1) * binomial(n-1,k-1)/k!.
a(n) ~ (3 + exp(-1))^(n + 1/2) * n^(n-1) / 3^(3/2). - Vaclav Kotesovec, Sep 10 2024