A073315 - OEIS

%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