%I #40 May 06 2023 09:00:42

%S 1,1,-1,1,0,1,1,1,3,-1,1,2,13,17,1,1,3,31,173,169,-1,1,4,57,629,3321,

%T 2079,1,1,5,91,1547,18025,81529,31261,-1,1,6,133,3089,58993,662639,

%U 2443333,554483,1,1,7,183,5417,147081,2888979,29752957,86475493,11336753,-1

%N Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) = (-1)^n * Sum_{j=0..n} (-k*j)^j * binomial(n,j).

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/LambertW-Function.html">Lambert W-Function</a>.

%F E.g.f. of column k: exp(-x) / (1 + LambertW(-k*x)).

%F G.f. of column k: Sum_{j>=0} (k*j*x)^j / (1 + x)^(j+1).

%e Square array begins:

%e 1, 1, 1, 1, 1, 1, ...

%e -1, 0, 1, 2, 3, 4, ...

%e 1, 3, 13, 31, 57, 91, ...

%e -1, 17, 173, 629, 1547, 3089, ...

%e 1, 169, 3321, 18025, 58993, 147081, ...

%e -1, 2079, 81529, 662639, 2888979, 8998399, ...

%o (PARI) T(n, k) = (-1)^n*sum(j=0, n, (-k*j)^j*binomial(n, j));

%Y Columns k=0..3 give A033999, (-1)^n * A069856(n), A362859, A362860.

%Y Main diagonal gives A362862.

%Y Cf. A362856.

%K sign,tabl

%O 0,9

%A _Seiichi Manyama_, May 05 2023