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E.g.f. satisfies A(x)^A(x) = 1/(1 - x)^x.
5

%I #22 Mar 04 2024 07:46:47

%S 1,0,2,3,-4,-30,294,3780,-7904,-444528,78840,99657360,539299848,

%T -27852945120,-361237078944,10124338180320,258341121976320,

%U -4020500134465920,-205187357182405824,1330097523844832640,186823640933648588160,500469438126120583680

%N E.g.f. satisfies A(x)^A(x) = 1/(1 - x)^x.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/LambertW-Function.html">Lambert W-Function</a>.

%F a(n) = n! * Sum_{k=0..floor(n/2)} (-k+1)^(k-1) * |Stirling1(n-k,k)|/(n-k)!.

%F E.g.f.: A(x) = Sum_{k>=0} (-k+1)^(k-1) * (-x * log(1-x))^k / k!.

%F E.g.f.: A(x) = exp( LambertW(-x * log(1-x)) ).

%F E.g.f.: A(x) = -x * log(1-x)/LambertW(-x * log(1-x)).

%t nmax = 21; A[_] = 1;

%t Do[A[x_] = (1/(1 - x)^x)^(1/A[x]) + O[x]^(nmax+1) // Normal, {nmax}];

%t CoefficientList[A[x], x]*Range[0, nmax]! (* _Jean-François Alcover_, Mar 04 2024 *)

%o (PARI) a(n) = n!*sum(k=0, n\2, (-k+1)^(k-1)*abs(stirling(n-k, k, 1))/(n-k)!);

%o (PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, (-k+1)^(k-1)*(-x*log(1-x))^k/k!)))

%o (PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(lambertw(-x*log(1-x)))))

%o (PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace(-x*log(1-x)/lambertw(-x*log(1-x))))

%Y Cf. A355842, A356795, A356796, A356906.

%Y Cf. A349561, A356884.

%K sign

%O 0,3

%A _Seiichi Manyama_, Sep 03 2022