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A360176 Triangle read by rows. T(n, k) = Sum_{j=k..n} binomial(n, j) * (-j)^(n - j) * (-1)^(j - k)* A360177(j, k).

%I A360176 #7 Jan 29 2023 21:02:26
%S A360176 1,0,1,0,-5,1,0,37,-15,1,0,-393,223,-30,1,0,5481,-3815,745,-50,1,0,
%T A360176 -95053,76051,-18870,1865,-75,1,0,1975821,-1749811,514381,-65730,3920,
%U A360176 -105,1,0,-47939601,45876335,-15316854,2358181,-183610,7322,-140,1
%N A360176 Triangle read by rows. T(n, k) = Sum_{j=k..n} binomial(n, j) * (-j)^(n - j) * (-1)^(j - k)* A360177(j, k).
%F A360176 E.g.f. of column k: (1 - exp(-LambertW(x*exp(-x))))^k / k!.
%e A360176 Triangle T(n, k) starts:
%e A360176 [0] 1;
%e A360176 [1] 0,         1;
%e A360176 [2] 0,        -5,        1;
%e A360176 [3] 0,        37,      -15,         1;
%e A360176 [4] 0,      -393,      223,       -30,       1;
%e A360176 [5] 0,      5481,    -3815,       745,     -50,       1;
%e A360176 [6] 0,    -95053,    76051,    -18870,    1865,     -75,    1;
%e A360176 [7] 0,   1975821, -1749811,    514381,  -65730,    3920, -105,    1;
%e A360176 [8] 0, -47939601, 45876335, -15316854, 2358181, -183610, 7322, -140, 1;
%p A360176 T := (n, k) -> add(binomial(n, j) * (-j)^(n - j) * (-1)^(j - k) * A360177(j, k), j = k..n): for n from 0 to 9 do seq(T(n, k), k = 0..n) od;
%p A360176 # Alternative:
%p A360176 egf := k -> (1 - exp(-LambertW(x*exp(-x))))^k / k!:
%p A360176 ser := k -> series(egf(k), x, 22): T := (n, k) -> n!*coeff(ser(k), x, n):
%p A360176 for n from 0 to 8 do seq(T(n, k), k = 0..n) od;
%Y A360176 Cf. A360177, A273954 (column 1), A028895 (subdiagonal).
%K A360176 sign,tabl
%O A360176 0,5
%A A360176 _Peter Luschny_, Jan 28 2023