A180680 Expansion of e.g.f. (1 - sqrt(1 - 4*LambertW(x)))/2.

1, 0, 9, 8, 705, 2304, 154945, 1097088, 67699233, 821657600, 49591224441, 901894072320, 54967757171041, 1372143152529408, 86054992196720625, 2772416395058315264, 181071792859314812865, 7195828128158173888512, 493215560390253143533033

Offset: 1

Author

Svetlana Khomich (hsl(AT)cyberline.ru), Sep 15 2010

Keywords

• nonn

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..200

Programs

Mathematica

Rest[CoefficientList[Series[(1-Sqrt[1-4*LambertW[x]])/2, {x, 0, 20}], x]* Range[0, 20]!] (* Vaclav Kotesovec, Nov 27 2012 *)

PARI

x='x+O('x^50); Vec(serlaplace((1 - sqrt(1 - 4*lambertw(x)))/2)) \\ G. C. Greubel, Nov 08 2017

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

Formula

a(n) = n!*Sum_{k=1..n} (-1)^(n-k)*n^(n-k-1)*binomial(2*(k-1),k-1)/(n-k)!.

a(n) ~ (4/exp(5/4))^n / sqrt(10) * n^(n-1). - Vaclav Kotesovec, Nov 27 2012

E.g.f.: LambertW(x)/(1 - LambertW(x)/(1 - LambertW(x)/(1 - LambertW(x)/(1 - ...)))), a continued fraction. - Ilya Gutkovskiy, Nov 19 2017