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A239230 Expansion of -x*log'(-sqrt(12*x+2*sqrt(1-4*x)+2)/4+sqrt(1-4*x)/4+5/4).

%I A239230 #37 Mar 04 2025 23:15:07
%S A239230 0,1,1,4,9,36,112,428,1505,5692,21026,79806,301488,1151866,4403778,
%T A239230 16929474,65204353,251947668,975366094,3784197606,14705937794,
%U A239230 57242631464,223121176224,870805992278,3402485053664,13308485156086,52104519751272,204176144516818
%N A239230 Expansion of -x*log'(-sqrt(12*x+2*sqrt(1-4*x)+2)/4+sqrt(1-4*x)/4+5/4).
%F A239230 G.f.: A(x) = x*F'(x)/(1-F(x)), where F(x) is g.f. of A055113.
%F A239230 a(n) = n * Sum_{k=1..n} (Sum_{j=0..n-k} C(n+2*j-1,j+n-1) * (-1)^(k+j+n) * C(2*n-k,j+n)) / (2*n-k).
%F A239230 a(n) = Sum_{k=1..n} (-1)^(k+n) * C(2*n-k-1,n-1) * hypergeom([k-n, n/2+1/2, n/2], [n, n+1], 4). - _Peter Luschny_, May 22 2014
%F A239230 Conjecture D-finite with recurrence +24*(365025561*n-1672569283)*(n-1)*(n-2)*(2*n-1)*a(n) -4*(n-2)*(27129169947*n^3-209577621466*n^2+463278020461*n-314084557758)*a(n-1) +2*(-28823853823*n^4+487259692534*n^3-3105214937957*n^2+8814274338098*n-9143920331436)*a(n-2) +4*(276083065830*n^4-4172118623320*n^3+24824880820695*n^2-69263721795041*n+75832154222148)*a(n-3) +(-587491214125*n^4+9941738070620*n^3-75070680472775*n^2+281912285021344*n-413197788157152)*a(n-4) +2*(-1186924847911*n^4+26108844767699*n^3-211936472383904*n^2+757584729548632*n-1009721693733312)*a(n-5) +4*(2*n-11)*(328530544924*n^3-5280431217363*n^2+28334632524947*n-50473913356356)*a(n-6) -72*(4143100547*n-18456753180)*(n-6)*(2*n-11)*(2*n-13)*a(n-7)=0. - _R. J. Mathar_, Jul 27 2022
%F A239230 Conjecture: a(n) = Sum_{i=0..n-1} A059260(2*(n-1),n+i-1)*A009766(n+i-1,i)*(-1)^(n+i-1) = n*Sum_{i=0..n-1} binomial(n+2*i, i)/(n+2*i)*(-1)^(n+i-1)*[x^(n+i-1)] (1+(x+1)^(2*n-1))/(x+2) for n >= 0. - _Mikhail Kurkov_, Feb 18 2025
%p A239230 a:= n-> add((-1)^(k+n)*binomial(2*n-k-1, n-1)*hypergeom([k-n, (n+1)/2, n/2], [n, n+1], 4), k=1..n);
%p A239230 seq(round(evalf(a(n), 32)), n=0..24); # _Peter Luschny_, May 22 2014
%o A239230 (Maxima)
%o A239230 a(n):=n*sum(sum(binomial(n+2*j-1,j+n-1)*(-1)^(k+j+n)*binomial(2*n-k,j+n), j,0,n-k)/(2*n-k), k,1,n);
%Y A239230 Cf. A055113.
%K A239230 nonn
%O A239230 0,4
%A A239230 _Vladimir Kruchinin_, May 22 2014