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A349654 E.g.f. satisfies: A(x)^2 * log(A(x)) = exp(x) - 1.

%I A349654 #25 Feb 16 2025 08:34:02
%S A349654 1,1,-2,17,-213,3712,-82773,2250565,-72218912,2671680015,
%T A349654 -111950278213,5240764049094,-271082407059027,15353947287972373,
%U A349654 -945097225235334538,62820021683240176445,-4484426869618973019249,342169496779859317566456
%N A349654 E.g.f. satisfies: A(x)^2 * log(A(x)) = exp(x) - 1.
%H A349654 Seiichi Manyama, <a href="/A349654/b349654.txt">Table of n, a(n) for n = 0..356</a>
%H A349654 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/LambertW-Function.html">Lambert W-Function</a>.
%F A349654 a(n) = Sum_{k=0..n} (-2*k+1)^(k-1) * Stirling2(n,k).
%F A349654 E.g.f.: A(x) = exp( LambertW(2*(exp(x) - 1))/2 ).
%F A349654 G.f.: Sum_{k>=0} (-2*k+1)^(k-1) * x^k/Product_{j=1..k} (1 - j*x).
%F A349654 a(n) ~ -(-1)^n * sqrt(2*exp(1) - 1) * sqrt(log(2) - log(2 - exp(-1))) * n^(n-1) / (2 * exp(n + 1/2) * (log(2) - log(2*exp(1) - 1) + 1)^n). - _Vaclav Kotesovec_, Nov 24 2021
%p A349654 b:= proc(n, m) option remember; `if`(n=0,
%p A349654      (1-2*m)^(m-1), m*b(n-1, m)+b(n-1, m+1))
%p A349654     end:
%p A349654 a:= n-> b(n, 0):
%p A349654 seq(a(n), n=0..21);  # _Alois P. Heinz_, Jul 29 2022
%t A349654 a[n_] := Sum[(-2*k + 1)^(k - 1) * StirlingS2[n, k], {k, 0, n}]; Array[a, 18, 0] (* _Amiram Eldar_, Nov 27 2021 *)
%o A349654 (PARI) a(n) = sum(k=0, n, (-2*k+1)^(k-1)*stirling(n, k, 2));
%o A349654 (PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(lambertw(2*(exp(x)-1))/2)))
%o A349654 (PARI) my(N=20, x='x+O('x^N)); Vec(sum(k=0, N, (-2*k+1)^(k-1)*x^k/prod(j=1, k, 1-j*x)))
%Y A349654 Cf. A349650, A349652, A349656.
%Y A349654 Cf. A008277, A349583, A349655.
%K A349654 sign
%O A349654 0,3
%A A349654 _Seiichi Manyama_, Nov 23 2021