A073315 - OEIS

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, 67885776, 68584320, 33022080, 6531840, 362880

(list; table; graph; refs; listen; history; text; internal format)

OFFSET

1,3

LINKS

Hugo Pfoertner, Table of n, a(n) for n = 1..10011, rows 1..141, flattened.

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. See also on Research Gate.

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, k) = k!*((-1)^(n-k)*Stirling1(n, k))*C(n+1, k)/(n+1). - Vladimir Kruchinin, Sep 21 2018

T(n, k) = T(n-1, k-1) * n + (T(n-1, k) * n * (n - 1)) / (n - k + 1) with T(n, n) = n!. - Peter Luschny, Nov 15 2025

EXAMPLE

Triangle begins:

[1] 1;

[2] 1, 2;

[3] 2, 9, 6;

[4] 6, 44, 72, 24;

[5] 24, 250, 700, 600, 120;

[6] 120, 1644, 6750, 10200, 5400, 720;

[7] 720, 12348, 68208, 154350, 147000, 52920, 5040;

[8] 5040, 104544, 735392, 2274384, 3292800, 2163840, 564480, 40320;

[9] 40320, 986256, 8504928, 33911136, 67885776, 68584320, 33022080, 6531840, 362880;

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 */

(Python)

from functools import cache

@cache

def T(n: int, k: int) -> int:

if k < 1 or k > n or n < 1: return 0

if n == 1 and k == 1: return 1

return T(n-1, k-1) * n + (T(n-1, k) * n * (n - 1)) // (n - k + 1)

for n in range(1, 10): print([T(n, k) for k in range(1, n+1)]) # Peter Luschny, Nov 15 2025

CROSSREFS

Cf. A042977, A013703, A000142 (diagonal), A138013 (row sums).