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A362856 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).

%I A362856 #17 Feb 16 2025 08:34:05
%S A362856 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 A362856 2079,46656,1,-5,19,-37,121,1373,31261,823543,1,-6,28,-83,208,797,
%U A362856 21028,554483,16777216,1,-7,39,-153,441,21,14517,373931,11336753,387420489
%N A362856 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 A362856 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/LambertW-Function.html">Lambert W-Function</a>.
%F A362856 E.g.f. of column k: exp(-k*x) / (1 + LambertW(-x)).
%F A362856 G.f. of column k: Sum_{j>=0} (j*x)^j / (1 + k*x)^(j+1).
%e A362856 Square array begins:
%e A362856      1,    1,    1,   1,   1,     1, ...
%e A362856      1,    0,   -1,  -2,  -3,    -4, ...
%e A362856      4,    3,    4,   7,  12,    19, ...
%e A362856     27,   17,    7,  -9, -37,   -83, ...
%e A362856    256,  169,  120, 121, 208,   441, ...
%e A362856   3125, 2079, 1373, 797,  21, -1525, ...
%o A362856 (PARI) T(n, k) = sum(j=0, n, (-k)^(n-j)*j^j*binomial(n,j));
%Y A362856 Columns k=0..3 give A000312, (-1)^n * A069856(n), A362857, A362858.
%Y A362856 Main diagonal gives A290158.
%Y A362856 Cf. A362019.
%K A362856 sign,tabl
%O A362856 0,6
%A A362856 _Seiichi Manyama_, May 05 2023