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A292866 a(n) = n! * [x^n] exp(n*(1 - exp(x))).

%I A292866 #34 Dec 23 2021 06:02:57
%S A292866 1,-1,2,-3,-20,370,-4074,34293,-138312,-2932533,106271090,-2192834490,
%T A292866 32208497124,-206343936097,-7657279887698,412496622532785,
%U A292866 -12455477719752976,260294034150380430,-2256541295745391542,-122593550603339550843,8728842979656718306780
%N A292866 a(n) = n! * [x^n] exp(n*(1 - exp(x))).
%H A292866 Alois P. Heinz, <a href="/A292866/b292866.txt">Table of n, a(n) for n = 0..415</a>
%F A292866 a(n) = exp(n) * Sum_{k>=0} (-n)^k*k^n/k!. - _Ilya Gutkovskiy_, Jul 13 2019
%F A292866 a(n) = Sum_{k=0..n} (-n)^k * Stirling2(n,k). - _Seiichi Manyama_, Jul 28 2019
%F A292866 a(n) = BellPolynomial(n, -n). - _Peter Luschny_, Dec 23 2021
%p A292866 b:= proc(n, k) option remember; `if`(n=0, 1,
%p A292866       -(1+add(binomial(n-1, j-1)*b(n-j, k), j=1..n-1))*k)
%p A292866     end:
%p A292866 a:= n-> b(n$2):
%p A292866 seq(a(n), n=0..30);  # _Alois P. Heinz_, Sep 25 2017
%t A292866 Table[n!*SeriesCoefficient[E^(n*(1 - E^x)),{x,0,n}], {n,0,20}] (* _Vaclav Kotesovec_, Sep 25 2017 *)
%t A292866 a[n_] := BellB[n, -n]; Table[a[n], {n, 0, 20}] (* _Peter Luschny_, Dec 23 2021 *)
%o A292866 (Ruby)
%o A292866 def ncr(n, r)
%o A292866   return 1 if r == 0
%o A292866   (n - r + 1..n).inject(:*) / (1..r).inject(:*)
%o A292866 end
%o A292866 def A(k, n)
%o A292866   ary = [1]
%o A292866   (1..n).each{|i| ary << k * (0..i - 1).inject(0){|s, j| s + ncr(i - 1, j) * ary[j]}}
%o A292866   ary
%o A292866 end
%o A292866 def A292866(n)
%o A292866   (0..n).map{|i| A(-i, i)[-1]}
%o A292866 end
%o A292866 p A292866(20)
%o A292866 (PARI) {a(n) = sum(k=0, n, (-n)^k*stirling(n, k, 2))} \\ _Seiichi Manyama_, Jul 28 2019
%Y A292866 Main diagonal of A292861.
%Y A292866 Cf. A242817.
%K A292866 sign
%O A292866 0,3
%A A292866 _Seiichi Manyama_, Sep 25 2017