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A320096 a(n) = Sum_{k=1..n} (-1)^(n-k) * Stirling1(n,k) * k! * k^n, with a(0)=1.

%I A320096 #23 Jun 04 2022 03:35:04
%S A320096 1,1,9,212,9418,675014,71092502,10334690232,1982433606264,
%T A320096 485065343565072,147433546709109408,54493722609862927632,
%U A320096 24069397682825072219040,12520250948941157091235344,7575515622713954399390221008,5275250174853125498317783254528
%N A320096 a(n) = Sum_{k=1..n} (-1)^(n-k) * Stirling1(n,k) * k! * k^n, with a(0)=1.
%H A320096 Seiichi Manyama, <a href="/A320096/b320096.txt">Table of n, a(n) for n = 0..229</a>
%F A320096 a(n) ~ c * d^n * n^(2*n + 1/2), where
%F A320096 w = -LambertW(-1, -exp(-r)*r) = 1.1628296650659469964248518258036278907318113...
%F A320096 r = 0.8531304407911771560472963194514988627832723535823134189532... is the real root of the equation w = r + exp(-1/r)
%F A320096 d = exp(-1)*r*w*(w-r)^(r-1) = 0.433513333588184444899487502412976956849408575992...
%F A320096 c = 1.959633090979666812031505093625147349925787002426082...
%F A320096 E.g.f.: Sum_{k>=0} (-log(1 - k*x))^k. - _Seiichi Manyama_, Feb 02 2022
%t A320096 Flatten[{1, Table[Sum[(-1)^(n-k)*StirlingS1[n, k]*k!*k^n, {k, 1, n}], {n, 1, 20}]}]
%t A320096 nmax = 20; CoefficientList[Series[1 + Sum[(-Log[1 - k*x])^k, {k, 1, nmax}], {x, 0, nmax}], x] * Range[0, nmax]! (* _Vaclav Kotesovec_, Jun 04 2022 *)
%o A320096 (PARI) a(n) = sum(k=0, n, (-1)^(n-k)*k!*k^n*stirling(n, k, 1)); \\ _Seiichi Manyama_, Feb 02 2022
%o A320096 (PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, (-log(1-k*x))^k))) \\ _Seiichi Manyama_, Feb 02 2022
%Y A320096 Cf. A007840, A220181, A320083.
%K A320096 nonn
%O A320096 0,3
%A A320096 _Vaclav Kotesovec_, Oct 05 2018