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A243585 Expansion of x*log'(C(C(x)-1)-1), C(x) = (1-sqrt(1-4*x))/(2*x).

%I A243585 #32 Nov 09 2024 07:49:13
%S A243585 1,4,20,106,580,3244,18446,106250,618340,3628600,21438820,127377980,
%T A243585 760346350,4556473276,27396081950,165189725326,998492094244,
%U A243585 6048338850560,36706629690824,223139239595840,1358475322091620
%N A243585 Expansion of x*log'(C(C(x)-1)-1), C(x) = (1-sqrt(1-4*x))/(2*x).
%H A243585 G.-S. Cheon, H. Kim, L. W. Shapiro, <a href="http://arxiv.org/abs/1410.1249">Mutation effects in ordered trees</a>, arXiv preprint arXiv:1410.1249 [math.CO], 2014 (see page 6).
%F A243585 a(n) = Sum_{k=0..n} binomial(2*k,k)*binomial(2*n,n-k).
%F A243585 a(n) ~ 5^(2*n+1/2) / (4^n * sqrt(3*Pi*n)). - _Vaclav Kotesovec_, Jun 08 2014
%F A243585 First column of A094527^2. 1 + x*exp( Sum_{n >= 1} a(n)*x^n/n ) = 1 + x + 4*x^2 + 18*x^3 + 86*x^4 + ... is the o.g.f. for A153294. - _Peter Bala_, Jul 21 2015
%F A243585 Conjecture D-finite with recurrence: 2*n*(2*n-1)*(3*n-5)*a(n) +(-123*n^3+328*n^2-249*n+60)*a(n-1) +50*(n-1)*(2*n-3)*(3*n-2)*a(n-2)=0. - _R. J. Mathar_, Jun 14 2016
%F A243585 a(n) = binomial(2*n, n)*hypergeom([1/2, -n], [n + 1], -4). - _Peter Luschny_, Aug 04 2019
%t A243585 CoefficientList[Series[1/(Sqrt[(1-4*x)*(2*Sqrt[1-4*x]+5*x-2)/x]), {x, 0, 20}], x] (* _Vaclav Kotesovec_, Jun 08 2014 *)
%t A243585 A243585[n_] := Binomial[2 n, n] Hypergeometric2F1[1/2, -n, n + 1, -4];
%t A243585 Table[A243585[n], {n, 0, 20}] (* _Peter Luschny_, Aug 04 2019 *)
%o A243585 (Maxima)
%o A243585 a(n):=sum(binomial(2*k,k)*binomial(2*n,n-k),k,0,n);
%Y A243585 Cf. A000108, A000984, A094527, A153294.
%K A243585 nonn
%O A243585 0,2
%A A243585 _Vladimir Kruchinin_, Jun 07 2014