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%I #10 Jan 29 2023 21:02:21
%S 1,0,1,0,3,1,0,16,9,1,0,125,91,18,1,0,1296,1105,295,30,1,0,16807,
%T 15961,5160,725,45,1,0,262144,269297,99631,17290,1505,63,1,0,4782969,
%U 5217031,2135070,431221,46970,2786,84,1,0,100000000,114358881,50631967,11477046,1471701,110250,4746,108,1
%N Triangle read by rows. T(n, k) = 1 if n = k, otherwise T(n, k) = Sum_{j=0..k-1} (-1)^(j - k - 1) * (n + j + 1)^(n-1) / (j! * (k - 1 - j)!).
%F T(n, k) = n! * [x^n] ((-LambertW(-x)/x - 1)^k / k!).
%e Triangle T(n, k) starts:
%e [0] 1;
%e [1] 0, 1;
%e [2] 0, 3, 1;
%e [3] 0, 16, 9, 1;
%e [4] 0, 125, 91, 18, 1;
%e [5] 0, 1296, 1105, 295, 30, 1;
%e [6] 0, 16807, 15961, 5160, 725, 45, 1;
%e [7] 0, 262144, 269297, 99631, 17290, 1505, 63, 1;
%e [8] 0, 4782969, 5217031, 2135070, 431221, 46970, 2786, 84, 1;
%p A360177 := (n, k) -> if n = k then 1 else
%p add((-1)^(u-k-1)*(n+u+1)^(n-1)/(u!*(k-1-u)!), u = 0.. k-1) fi:
%p for n from 0 to 8 do seq(A360177(n, k), k = 0..n) od;
%p # Alternative:
%p egf := k -> (-LambertW(-x)/x - 1)^k / k!:
%p ser := k -> series(egf(k), x, 22): T := (n, k) -> n!*coeff(ser(k), x, n):
%p for n from 0 to 8 do print(seq(T(n, k), k = 0..n)) od;
%Y Cf. A124824 (row sums), A000272 (column 1), A045943 (subdiagonal).
%K nonn,tabl
%O 0,5
%A _Peter Luschny_, Jan 28 2023
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