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%I #7 Apr 18 2020 11:50:02
%S 1,0,-2,-9,-16,625,21384,571438,13471744,188661555,-9794500000,
%T -1476328587789,-134710712340480,-10664210861777200,
%U -744650964057237888,-37832162051689453125,831929248561267474432,725944099523076464203157,167435684777981700601449984
%N a(0) = 1; thereafter a(n) = exp(1/n) * Sum_{k>=0} (n*k + 1)^n / ((-n)^k * k!).
%F a(n) = [x^n] (1/(1 - x)) * Sum_{k>=0} (-x/(1 - x))^k / Product_{j=1..k} (1 - n*j*x/(1 - x)).
%F a(n) = n! * [x^n] exp(x + (1 - exp(n*x)) / n), for n > 0.
%F a(n) = A334192(n,n).
%t Table[SeriesCoefficient[1/(1 - x) Sum[(-x/(1 - x))^k/Product[(1 - n j x/(1 - x)), {j, 1, k}], {k, 0, n}], {x, 0, n}], {n, 0, 18}]
%t Join[{1}, Table[n! SeriesCoefficient[Exp[x + (1 - Exp[n x])/n], {x, 0, n}], {n, 1, 18}]]
%t Join[{1}, Table[Sum[Binomial[n, k]*n^k*BellB[k, -1/n], {k, 0, n}], {n, 1, 18}]] (* _Vaclav Kotesovec_, Apr 18 2020 *)
%Y Cf. A318183, A334162, A334190, A334191, A334192.
%K sign
%O 0,3
%A _Ilya Gutkovskiy_, Apr 18 2020
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