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A349720 E.g.f. satisfies: A(x) = exp( x * (1 + 1/A(x)^2)/2 ).

%I A349720 #23 Feb 16 2025 08:34:02
%S A349720 1,1,-1,7,-63,801,-13025,258343,-6048511,163276417,-4992740289,
%T A349720 170571634311,-6439161507647,266180947507489,-11958385377911713,
%U A349720 580151397382158631,-30227616424300542975,1683438461080186841601,-99796591057813372007297
%N A349720 E.g.f. satisfies: A(x) = exp( x * (1 + 1/A(x)^2)/2 ).
%H A349720 Seiichi Manyama, <a href="/A349720/b349720.txt">Table of n, a(n) for n = 0..372</a>
%H A349720 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/LambertW-Function.html">Lambert W-Function</a>.
%F A349720 a(n) = (1/2^n) * Sum_{k=0..n} (-2*k+1)^(n-1) * binomial(n,k).
%F A349720 E.g.f.: ( x/LambertW( x * exp(-x) ) )^(1/2).
%F A349720 G.f.: 2 * Sum_{k>=0} (-2*k+1)^(k-1) * x^k/(2 - (-2*k+1)*x)^(k+1).
%F A349720 a(n) ~ -(-1)^n * sqrt(1 + LambertW(exp(-1))) * n^(n-1) / (2 * exp(n) * LambertW(exp(-1))^(n - 1/2)). - _Vaclav Kotesovec_, Dec 05 2021
%t A349720 a[n_] := (1/2^n) * Sum[(-2*k + 1)^(n - 1) * Binomial[n, k], {k, 0, n}]; Array[a, 19, 0] (* _Amiram Eldar_, Nov 27 2021 *)
%o A349720 (PARI) a(n) = sum(k=0, n, (-2*k+1)^(n-1)*binomial(n, k))/2^n;
%o A349720 (PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace((x/lambertw(x*exp(-x)))^(1/2)))
%o A349720 (PARI) my(N=20, x='x+O('x^N)); Vec(2*sum(k=0, N, (-2*k+1)^(k-1)*x^k/(2-(-2*k+1)*x)^(k+1)))
%Y A349720 Cf. A007889, A202617, A349714, A349715, A349716, A349719, A349721.
%K A349720 sign
%O A349720 0,4
%A A349720 _Seiichi Manyama_, Nov 27 2021