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%I #24 Mar 19 2024 09:34:41
%S 1,0,9,8,705,2304,154945,1097088,67699233,821657600,49591224441,
%T 901894072320,54967757171041,1372143152529408,86054992196720625,
%U 2772416395058315264,181071792859314812865,7195828128158173888512,493215560390253143533033
%N Expansion of e.g.f. (1 - sqrt(1 - 4*LambertW(x)))/2.
%H Vincenzo Librandi, <a href="/A180680/b180680.txt">Table of n, a(n) for n = 1..200</a>
%F a(n) = n!*Sum_{k=1..n} (-1)^(n-k)*n^(n-k-1)*binomial(2*(k-1),k-1)/(n-k)!.
%F a(n) ~ (4/exp(5/4))^n / sqrt(10) * n^(n-1). - _Vaclav Kotesovec_, Nov 27 2012
%F E.g.f.: LambertW(x)/(1 - LambertW(x)/(1 - LambertW(x)/(1 - LambertW(x)/(1 - ...)))), a continued fraction. - _Ilya Gutkovskiy_, Nov 19 2017
%t Rest[CoefficientList[Series[(1-Sqrt[1-4*LambertW[x]])/2, {x, 0, 20}], x]* Range[0, 20]!] (* _Vaclav Kotesovec_, Nov 27 2012 *)
%o (PARI) x='x+O('x^50); Vec(serlaplace((1 - sqrt(1 - 4*lambertw(x)))/2)) \\ _G. C. Greubel_, Nov 08 2017
%o (PARI) a(n) = n!*sum(k=1, n, (-1)^(n-k)*n^(n-k-1)*binomial(2*(k-1),k-1)/(n-k)!); \\ _Michel Marcus_, Nov 09 2017
%K nonn
%O 1,3
%A Svetlana Khomich (hsl(AT)cyberline.ru), Sep 15 2010