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%I #10 May 06 2023 09:00:27
%S 1,0,13,629,58993,8998399,2035844461,640881617123,267995012680641,
%T 143734541641235567,96200314049944377901,78599287990433271805699,
%U 76993408916168689318057201,89072357257840197226050646151
%N a(n) = (-1)^n * Sum_{k=0..n} (-n*k)^k * binomial(n,k).
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/LambertW-Function.html">Lambert W-Function</a>.
%F a(n) = n! * [x^n] exp(-x) / (1 + LambertW(-n*x)).
%F a(n) = [x^n] Sum_{k>=0} (n*k*x)^k / (1 + x)^(k+1).
%o (PARI) a(n) = (-1)^n * sum(k=0, n, (-n*k)^k*binomial(n, k));
%Y Main diagonal of A362019.
%Y Cf. A290158.
%K nonn
%O 0,3
%A _Seiichi Manyama_, May 06 2023