A396675 Square array A(n,k), n>=1, k>=0, read by antidiagonals downwards, where A(n,k) = n! * [x^n] log(1 - W_k(-x)) and W_k(x) is the k-th iterate of LambertW(x).

1, 1, -1, 1, 1, 2, 1, 3, 5, -6, 1, 5, 20, 34, 24, 1, 7, 47, 206, 329, -120, 1, 9, 86, 654, 2884, 4056, 720, 1, 11, 137, 1522, 12129, 51222, 60997, -5040, 1, 13, 200, 2954, 35384, 282318, 1104970, 1082320, 40320, 1, 15, 275, 5094, 82849, 1023324, 7929669, 28092136, 22137201, -362880

Offset

1,6

Links

Table of n, a(n) for n=1..55.

Formula

A(n,0) = (-1)^(n-1) * (n-1)!; A(n,k) = Sum_{j=1..n} n^(n-j) * binomial(n-1,j-1) * A(j,k-1) for k > 0.

A(n,k) = (n-1)! * Sum_{x_1, x_2, ..., x_{k+1} >= 0 and x_1 + x_2 + ... + x_{k+1} = n-1} (-1)^(x_{k+1}) * Product_{i=1..k} ((n - Sum_{j=1..i-1} x_j)^(x_i) / x_i!).

Example

Square array begins:
     1,    1,     1,      1,       1,       1,       1, ...
    -1,    1,     3,      5,       7,       9,      11, ...
     2,    5,    20,     47,      86,     137,     200, ...
    -6,   34,   206,    654,    1522,    2954,    5094, ...
    24,  329,  2884,  12129,   35384,   82849,  167604, ...
  -120, 4056, 51222, 282318, 1023324, 2871660, 6782586, ...
  ...

Prog

(PARI) a(n, k) = if(k==0, (-1)^(n-1)*(n-1)!, sum(j=1, n, n^(n-j)*binomial(n-1, j-1)*a(j, k-1)));

Crossrefs

Columns k=0..3 give A133942(n-1), A133297, A396677, A396678.

Cf. A274391, A396498, A396676.

Sequence in context: A179382 A161169 A239738 * A058202 A327452 A257982

Adjacent sequences: A396672 A396673 A396674 * A396676 A396677 A396678

Keyword

sign,tabl

Author

Seiichi Manyama, Jun 02 2026

Status

approved