%I #16 Sep 22 2018 17:05:53
%S 1,1,2,2,9,6,6,44,72,24,24,250,700,600,120,120,1644,6750,10200,5400,
%T 720,720,12348,68208,154350,147000,52920,5040,5040,104544,735392,
%U 2274384,3292800,2163840,564480,40320,40320,986256,8504928,33911136
%N Expansion of Lambert W function in powers of log(log(x))/log(x).
%H R. M. Corless, G. H. Gonnet, D. E. G. Hare, D. J. Jeffrey, and D. E. Knuth, <a href="http://www.apmaths.uwo.ca/~rcorless/frames/PAPERS/LambertW/LambertW.ps">On the Lambert W Function</a>, Advances in Computational Mathematics, (5), 1996, pp. 329-359.
%F 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!.
%F T(n,m) = m!*((-1)^(n-m)*Stirling1(n,m))*C(n+1,m)/(n+1). - _Vladimir Kruchinin_, Sep 21 2018
%e Triangle begins:
%e { 1}
%e { 1, 2}
%e { 2, 9, 6}
%e { 6, 44, 72, 24}
%e { 24, 250, 700, 600, 120}
%e {120, 1644, 6750, 10200, 5400, 720}
%e ...
%p T := (n, k) -> (-1)^(n-k)*Stirling1(n, k)*pochhammer(n-k+2, k-1):
%p for n from 1 to 6 do seq(T(n,k), k=1..n) od; # _Peter Luschny_, Sep 22 2018
%o (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))}
%o (Maxima)
%o T(n,m):=m!*((-1)^(n-m)*stirling1(n,m))*binomial(n+1,m)/(n+1); /* _Vladimir Kruchinin_, Sep 21 2018 */
%K nonn,tabl
%O 1,3
%A _Michael Somos_, Jul 24 2002