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A357247 E.g.f. satisfies A(x) * log(A(x)) = x * exp(-x).

%I A357247 #28 Feb 16 2025 08:34:04
%S A357247 1,1,-3,13,-103,1241,-19691,384805,-8918351,238966705,-7265920339,
%T A357247 247123552061,-9295263915191,383095792217737,-17167554097899323,
%U A357247 831082449069928021,-43221681697593767071,2403219105771778162529,-142263939562414917333155
%N A357247 E.g.f. satisfies A(x) * log(A(x)) = x * exp(-x).
%H A357247 Seiichi Manyama, <a href="/A357247/b357247.txt">Table of n, a(n) for n = 0..372</a>
%H A357247 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/LambertW-Function.html">Lambert W-Function</a>.
%F A357247 E.g.f. satisfies A(x) * log(A(x)) - x * exp(-x) = 0.
%F A357247 E.g.f.: A(x) = Sum_{k>=0} (-k+1)^(k-1) * (x * exp(-x))^k / k!.
%F A357247 E.g.f.: A(x) = exp( LambertW(x * exp(-x)) ).
%F A357247 E.g.f.: A(x) = x * exp(-x)/LambertW(x * exp(-x)).
%F A357247 a(0) = 1; a(n) = Sum_{k=1..n} (-1)^(k-1) * A216857(k) * binomial(n-1,k-1) * a(n-k).
%F A357247 a(n) = (-1)^(n - 1) * Sum_{j=0..n} binomial(n, j) * (j - 1)^(j - 1) * j^(n - j). - _Peter Luschny_, Jan 28 2023
%F A357247 a(n) ~ -(-1)^n * sqrt(1 + LambertW(exp(-1))) * n^(n-1) / (exp(n+1) * LambertW(exp(-1))^n). - _Vaclav Kotesovec_, Jan 28 2023
%p A357247 A357247 := n -> (-1)^(n - 1) * add(binomial(n, j) * (j - 1)^(j - 1) * j^(n - j), j = 0..n): seq(A357247(n), n = 0..18); # _Peter Luschny_, Jan 28 2023
%t A357247 nmax = 20; A[_] = 1;
%t A357247 Do[A[x_] = Exp[x/(Exp[x]*A[x])] + O[x]^(nmax+1) // Normal, {nmax}];
%t A357247 CoefficientList[A[x], x]*Range[0, nmax]! (* _Jean-François Alcover_, Mar 04 2024 *)
%o A357247 (PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, (-k+1)^(k-1)*(x*exp(-x))^k/k!)))
%o A357247 (PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(lambertw(x*exp(-x)))))
%o A357247 (PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace(x*exp(-x)/lambertw(x*exp(-x))))
%o A357247 (PARI) a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=1, i, sum(k=1, j, (-k)^(j-1)*binomial(j, k))*binomial(i-1, j-1)*v[i-j+1])); v;
%Y A357247 Cf. A177885, A216857, A357243, A357246, A359759 (column 1).
%K A357247 sign
%O A357247 0,3
%A A357247 _Seiichi Manyama_, Sep 19 2022