%I #22 Sep 09 2024 09:33:28

%S 1,1,8,64,824,12968,252720,5789712,153169440,4589004192,153643615872,

%T 5684390364288,230307823878144,10141452865049088,482259966649655808,

%U 24630247225278881280,1344614199041549399040,78137673004382654223360

%N Expansion of (1-sqrt(1-4*log(1+x)))/2.

%H G. C. Greubel, <a href="/A087138/b087138.txt">Table of n, a(n) for n = 1..372</a>

%H <a href="/index/Res#revert">Index entries for reversions of series</a>

%F a(n) = Sum_{k=1..n} Stirling1(n, k)*k!*Catalan(k-1).

%F a(n) ~ n! / (2*exp(1/8)*sqrt(Pi) * (exp(1/4)-1)^(n-1/2) * n^(3/2)). - _Vaclav Kotesovec_, May 03 2015

%F From _Seiichi Manyama_, Sep 09 2024: (Start)

%F E.g.f. satisfies A(x) = (log(1 + x)) / (1 - A(x)).

%F E.g.f.: Series_Reversion( exp(x * (1 - x)) - 1 ). (End)

%t Rest[CoefficientList[Series[(1-Sqrt[1-4*Log[1+x]])/2, {x, 0, 20}], x] * Range[0, 20]!] (* _Vaclav Kotesovec_, May 03 2015 *)

%o (PARI) x='x+O('x^50); Vec(serlaplace((1-sqrt(1-4*log(1+x)))/2)) \\ _G. C. Greubel_, May 24 2017

%Y Cf. A000108, A052851, A052803, A052886, A052895, A006531, A048287, A293604.

%Y Cf. A370462, A370463.

%K nonn

%O 1,3

%A _Vladeta Jovovic_, Oct 18 2003