A073315 Expansion of Lambert W function in powers of log(log(x))/log(x).

1, 1, 2, 2, 9, 6, 6, 44, 72, 24, 24, 250, 700, 600, 120, 120, 1644, 6750, 10200, 5400, 720, 720, 12348, 68208, 154350, 147000, 52920, 5040, 5040, 104544, 735392, 2274384, 3292800, 2163840, 564480, 40320, 40320, 986256, 8504928, 33911136

Offset

1,3

Links

Table of n, a(n) for n=1..40.

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

Sequence in context: A178236 A093589 A319129 * A298597 A066320 A005168

Adjacent sequences: A073312 A073313 A073314 * A073316 A073317 A073318

Keyword

nonn,tabl

Author

Michael Somos, Jul 24 2002

Status

approved