OFFSET
0,2
COMMENTS
The n-th derivative of W(x) at x=exp(1) is exp(-n) * a(n) / 2^(2*n). - Paolo Bonzini, Jun 23 2016
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..150
FORMULA
E.g.f.: LambertW(exp(1)*(1+4*x)). - Vladeta Jovovic, Nov 19 2003
|a(n)| ~ 4^n * exp(n) * n^(n-1) / (1+exp(2))^(n-1/2). - Vaclav Kotesovec, Jul 09 2013
EXAMPLE
LambertW(exp(1)*(1+4*x)) = 1 + 1/2*x - 3/16*x^2 + 19/192*x^3 - 185/3072*x^4 + ...
MAPLE
LambertW( exp(1)*(1+x) );
seq(n!*coeff(series(LambertW(exp(1)*(1+4*x)), x, n+1), x, n), n=0..20); # Vaclav Kotesovec, Jul 09 2013
MATHEMATICA
max = 17; (Series[ ProductLog[E*(1 + 4*x)], {x, 0, max}] // CoefficientList[#, x] &)*Range[0, max]! (* Jean-François Alcover, Jun 20 2013 *)
PROG
(Maxima)
a(n):= if n<2 then n+1 else 2*(n-1)!*(sum((binomial(n+k-1, n-1)*sum(binomial(k, j)*2^(n-j-1)*sum(binomial(j, l)*(-1)^(l)*sum((l^(n+j-i-1)*binomial(l, i))/(n+j-i-1)!, i, 0, l), l, 1, j), j, 1, k)), k, 1, n-1)); /* based on A042977, Paolo Bonzini, Jun 23 2016 */
(PARI) x='x+O('x^50); Vec(serlaplace(lambertw(exp(1)*(1+4*x)))) \\ G. C. Greubel, Nov 15 2017
CROSSREFS
KEYWORD
sign
AUTHOR
Robert Corless (rmc(AT)pineapple.apmaths.uwo.ca)
EXTENSIONS
More terms from N. J. A. Sloane.
More terms from Vincenzo Librandi, Jul 25 2013
STATUS
approved