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A360177 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)!).

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