A052803 - OEIS

%I #34 Jun 27 2022 08:36:38

%S 1,1,5,44,566,9674,207166,5343456,161405016,5591409720,218592034584,

%T 9521490534720,457329182411856,24014921905589328,1368772939062117936,

%U 84161443919543331840,5553011951023694408064,391360838810043628416384,29342876851060951124158848

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

%C Previous name was: A simple grammar.

%H Seiichi Manyama, <a href="/A052803/b052803.txt">Table of n, a(n) for n = 0..360</a>

%H INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=762">Encyclopedia of Combinatorial Structures 762</a>

%F E.g.f.: (1/2)/log(-1/(-1+x))*(1-(1-4*log(-1/(-1+x)))^(1/2)).

%F a(n) ~ 2*sqrt(2) * n^(n-1) / (exp(3*n/4) * (exp(1/4)-1)^(n-1/2)). - _Vaclav Kotesovec_, Sep 30 2013

%F a(n) = Sum_{k=0..n} (2k)!/(k+1)! * |Stirling1(n,k)|. - _Michael D. Weiner_, Dec 23 2014

%F E.g.f.: 1/(1 + log(1-x)/(1 + log(1-x)/(1 + log(1-x)/(1 + log(1-x)/(1 + ...))))), a continued fraction. - _Ilya Gutkovskiy_, Nov 19 2017

%p spec := [S,{C=Cycle(Z),S=Sequence(B),B=Prod(C,S)},labeled]: seq(combstruct[count](spec,size=n), n=0..20);

%t CoefficientList[Series[-1/(2*Log[1-x]) * (1-(1+4*Log[1-x])^(1/2)), {x, 0, 20}], x]* Range[0, 20]! (* _Vaclav Kotesovec_, Sep 30 2013 *)

%Y Cf. A006531, A086662, A087152.

%K easy,nonn

%O 0,3

%A encyclopedia(AT)pommard.inria.fr, Jan 25 2000

%E New name using e.g.f., _Vaclav Kotesovec_, Sep 30 2013