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E.g.f. satisfies A(x) = exp(x + x / A(x)^2).
4

%I #19 Feb 16 2025 08:34:05

%S 1,2,-4,56,-1008,25632,-833600,33067904,-1548418816,83597525504,

%T -5112566055936,349330707068928,-26374805535322112,

%U 2180554321981349888,-195926186031705505792,19010400989418574020608,-1980997069982960384409600,220651645970702249702326272

%N E.g.f. satisfies A(x) = exp(x + x / A(x)^2).

%H Seiichi Manyama, <a href="/A362735/b362735.txt">Table of n, a(n) for n = 0..339</a>

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

%F E.g.f.: sqrt( 2*x / LambertW(2*x*exp(-2*x)) ) = exp( x + LambertW(2*x*exp(-2*x))/2 ).

%F a(n) = Sum_{k=0..n} (-2*k+1)^(n-1) * binomial(n,k) = 2^n * A349720(n).

%o (PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(x+lambertw(2*x*exp(-2*x))/2)))

%Y Cf. A349562, A362693, A362694, A362734.

%Y Cf. A349720.

%K sign,changed

%O 0,2

%A _Seiichi Manyama_, May 01 2023