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a(n) = Sum_{k=0..n} Stirling1(n,k) * k! * (-n)^(n-k).
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%I #17 Jun 07 2022 10:55:43

%S 1,1,4,42,904,34070,2019888,174588120,20804747136,3276218158560,

%T 659664288364800,165425062846302336,50574549124825998336,

%U 18520126461205806360144,8003819275469728355033088,4031020344281171589447408000,2340375822778055527109749211136

%N a(n) = Sum_{k=0..n} Stirling1(n,k) * k! * (-n)^(n-k).

%H Seiichi Manyama, <a href="/A352074/b352074.txt">Table of n, a(n) for n = 0..233</a>

%F a(n) = n! * [x^n] 1 / (1 + log(1 - n*x) / n) for n > 0.

%F a(n) ~ n! * n^(n-2) * (1 + 2*log(n)/n). - _Vaclav Kotesovec_, Mar 03 2022

%t Unprotect[Power]; 0^0 = 1; Table[Sum[StirlingS1[n, k] k! (-n)^(n - k), {k, 0, n}], {n, 0, 16}]

%t Join[{1}, Table[n! SeriesCoefficient[1/(1 + Log[1 - n x]/n), {x, 0, n}], {n, 1, 16}]]

%o (PARI) a(n) = sum(k=0, n, stirling(n, k, 1)*k!*(-n)^(n-k)); \\ _Michel Marcus_, Mar 02 2022

%Y Cf. A007840, A092985, A094420, A227917, A317171, A317172, A331690, A352069, A352071.

%Y Cf. A081048.

%K nonn

%O 0,3

%A _Ilya Gutkovskiy_, Mar 02 2022