OFFSET
1,3
LINKS
R. M. Corless, G. H. Gonnet, D. E. G. Hare, D. J. Jeffrey, and D. E. Knuth, On the Lambert W Function, Advances in Computational Mathematics, (5), 1996, pp. 329-359.
FORMULA
E.g.f.: LambertW(x) = Sum_{n>0, k>=0} T(n, k)(-1/log(log(x)))^k(log(log(x))/log(x))^n/n!.
T(n,m) = m!*((-1)^(n-m)*Stirling1(n,m))*C(n+1,m)/(n+1). - Vladimir Kruchinin, Sep 21 2018
EXAMPLE
Triangle begins:
{ 1}
{ 1, 2}
{ 2, 9, 6}
{ 6, 44, 72, 24}
{ 24, 250, 700, 600, 120}
{120, 1644, 6750, 10200, 5400, 720}
...
MAPLE
T := (n, k) -> (-1)^(n-k)*Stirling1(n, k)*pochhammer(n-k+2, k-1):
for n from 1 to 6 do seq(T(n, k), k=1..n) od; # Peter Luschny, Sep 22 2018
PROG
(PARI) {T(n, k) = local(z, y); if( k<0 || k>=n, 0, z = O(x); y = 'y; for( i=1, n+1, z = -log(1 - x - x * y *z)); n! * polcoeff( polcoeff(z, n, x), k, y))}
(Maxima)
T(n, m):=m!*((-1)^(n-m)*stirling1(n, m))*binomial(n+1, m)/(n+1); /* Vladimir Kruchinin, Sep 21 2018 */
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Michael Somos, Jul 24 2002
STATUS
approved