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A349561 E.g.f. satisfies: A(x)^A(x) = 1/(1 - x).

%I A349561 #30 Feb 16 2025 08:34:02
%S A349561 1,1,0,3,-8,100,-834,11438,-159928,2762352,-52322160,1124320032,
%T A349561 -26509832040,686751503568,-19306448087640,586539826169880,
%U A349561 -19131996548499264,667157522614934016,-24762890955027112128,974824890777753840576,-40566428716555791936000
%N A349561 E.g.f. satisfies: A(x)^A(x) = 1/(1 - x).
%H A349561 Seiichi Manyama, <a href="/A349561/b349561.txt">Table of n, a(n) for n = 0..398</a>
%H A349561 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/LambertW-Function.html">Lambert W-Function</a>.
%F A349561 a(n) = (-1)^(n-1) * Sum_{k=0..n} (k-1)^(k-1) * Stirling1(n,k).
%F A349561 E.g.f. A(x) = -Sum_{k>=0} (k-1)^(k-1) * (log(1-x))^k / k!.
%F A349561 E.g.f.: A(x) = -log(1-x)/LambertW(-log(1-x)).
%F A349561 a(n) ~ -(-1)^n * n^(n-1) / ((exp(exp(-1)) - 1)^(n - 1/2) * exp(n + exp(-1)/2 + 1/2)). - _Vaclav Kotesovec_, Nov 22 2021
%e A349561 A(x) - 1 = x + 3*x^3/6 - 8*x^4/24 + ... = x + x^3/2 - x^4/3 + ... .
%e A349561 A(x)^A(x) = (1 + (A(x) - 1))^(1 + (A(x) - 1)) = Sum_{k>=0} A005727(k) * (A(x) - 1)^k / k! = 1 + 1 * (x + x^3/2 - x^4/3 + ... )/1! + 2 * (x + x^3/2 - x^4/3 + ... )^2/2! + 3 * (x + x^3/2 - x^4/3 + ... )^3/3! + ...  = 1 + x + x^2 + x^3 + ... = 1/(1 - x).
%t A349561 Join[{1}, Table[(n-1)! - (-1)^n*Sum[(k-1)^(k-1)*StirlingS1[n, k], {k, 2, n}], {n, 1, 20}]] (* _Vaclav Kotesovec_, Nov 22 2021 *)
%o A349561 (PARI) a(n) = (-1)^(n-1)*sum(k=0, n, (k-1)^(k-1)*stirling(n, k, 1));
%o A349561 (PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace(-sum(k=0, N, (k-1)^(k-1)*log(1-x)^k/k!)))
%o A349561 (PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace(-log(1-x)/lambertw(-log(1-x))))
%Y A349561 Cf. A005727, A120980, A141209, A216135, A216136, A229237, A305819.
%K A349561 sign
%O A349561 0,4
%A A349561 _Seiichi Manyama_, Nov 22 2021