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A362019 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).

%I A362019 #41 Feb 16 2025 08:34:05
%S A362019 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 A362019 2079,1,1,5,91,1547,18025,81529,31261,-1,1,6,133,3089,58993,662639,
%U A362019 2443333,554483,1,1,7,183,5417,147081,2888979,29752957,86475493,11336753,-1
%N A362019 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 A362019 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/LambertW-Function.html">Lambert W-Function</a>.
%F A362019 E.g.f. of column k: exp(-x) / (1 + LambertW(-k*x)).
%F A362019 G.f. of column k: Sum_{j>=0} (k*j*x)^j / (1 + x)^(j+1).
%e A362019 Square array begins:
%e A362019    1,    1,     1,      1,       1,       1, ...
%e A362019   -1,    0,     1,      2,       3,       4, ...
%e A362019    1,    3,    13,     31,      57,      91, ...
%e A362019   -1,   17,   173,    629,    1547,    3089, ...
%e A362019    1,  169,  3321,  18025,   58993,  147081, ...
%e A362019   -1, 2079, 81529, 662639, 2888979, 8998399, ...
%o A362019 (PARI) T(n, k) = (-1)^n*sum(j=0, n, (-k*j)^j*binomial(n, j));
%Y A362019 Columns k=0..3 give A033999, (-1)^n * A069856(n), A362859, A362860.
%Y A362019 Main diagonal gives A362862.
%Y A362019 Cf. A362856.
%K A362019 sign,tabl
%O A362019 0,9
%A A362019 _Seiichi Manyama_, May 05 2023