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Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) = Sum_{j=0..n} (-k)^(n-j) * j^j * binomial(n,j).
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%I #16 May 06 2023 09:00:48

%S 1,1,1,1,0,4,1,-1,3,27,1,-2,4,17,256,1,-3,7,7,169,3125,1,-4,12,-9,120,

%T 2079,46656,1,-5,19,-37,121,1373,31261,823543,1,-6,28,-83,208,797,

%U 21028,554483,16777216,1,-7,39,-153,441,21,14517,373931,11336753,387420489

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

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

%e Square array begins:

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

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

%e 4, 3, 4, 7, 12, 19, ...

%e 27, 17, 7, -9, -37, -83, ...

%e 256, 169, 120, 121, 208, 441, ...

%e 3125, 2079, 1373, 797, 21, -1525, ...

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

%Y Columns k=0..3 give A000312, (-1)^n * A069856(n), A362857, A362858.

%Y Main diagonal gives A290158.

%Y Cf. A362019.

%K sign,tabl

%O 0,6

%A _Seiichi Manyama_, May 05 2023