%I #19 Apr 21 2023 11:23:44
%S 1,1,1,1,1,1,1,1,0,1,1,1,-1,-5,1,1,1,-2,-11,-14,1,1,1,-3,-17,-11,56,1,
%T 1,1,-4,-23,10,381,736,1,1,1,-5,-29,49,976,2461,1114,1,1,1,-6,-35,106,
%U 1841,3736,-21083,-45156,1,1,1,-7,-41,181,2976,3121,-106910,-449623,-428660,1
%N Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) = n! * Sum_{j=0..floor(n/2)} (-k/2)^j * (j+1)^(n-j-1) / (j! * (n-2*j)!).
%H Seiichi Manyama, <a href="/A362394/b362394.txt">Antidiagonals n = 0..139, flattened</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/LambertW-Function.html">Lambert W-Function</a>.
%F E.g.f. A_k(x) of column k satisfies A_k(x) = exp(x - k*x^2/2 * A_k(x)).
%F A_k(x) = exp(x - LambertW(k*x^2/2 * exp(x))).
%F A_k(x) = 2 * LambertW(k*x^2/2 * exp(x))/(k*x^2) for k > 0.
%e Square array begins:
%e 1, 1, 1, 1, 1, 1, 1, ...
%e 1, 1, 1, 1, 1, 1, 1, ...
%e 1, 0, -1, -2, -3, -4, -5, ...
%e 1, -5, -11, -17, -23, -29, -35, ...
%e 1, -14, -11, 10, 49, 106, 181, ...
%e 1, 56, 381, 976, 1841, 2976, 4381, ...
%e 1, 736, 2461, 3736, 3121, -824, -9539, ...
%o (PARI) T(n, k) = n! * sum(j=0, n\2, (-k/2)^j*(j+1)^(n-j-1)/(j!*(n-2*j)!));
%Y Columns k=0..3 give A000012, A362395, A362396, A362397.
%Y Cf. A362277, A362377.
%K sign,tabl
%O 0,14
%A _Seiichi Manyama_, Apr 20 2023