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%I #29 Nov 11 2024 22:24:06
%S 1,5,59,782,10915,156890,2298254,34115772,511402275,7723927970,
%T 117355941274,1791748546692,27465854168974,422452379203652,
%U 6516524753922620,100771332997219832,1561717224800526627,24249283134262469490,377165907419518984802,5875218357778513634100
%N Expansion of 4*x/(16*x+(sqrt(2)*sqrt(sqrt(1-16*x)+1)-1)*sqrt(1-16*x)-1).
%H G. C. Greubel, <a href="/A249519/b249519.txt">Table of n, a(n) for n = 0..830</a>
%F G.f.: 4*x/(16*x+(sqrt(2)*sqrt(sqrt(1-16*x)+1)-1)*sqrt(1-16*x)-1).
%F a(n) = Sum_{i = 0..n} 2^i*binomial(2*n-i-1,n-i)*binomial(2*n+i-1,i).
%F a(n) ~ (1+sqrt(2)) * 2^(4*n-2) / sqrt(Pi*n). - _Vaclav Kotesovec_, Oct 31 2014
%F D-finite with recurrence: n*(2*n-1)*(n-1)*a(n) -2*(n-1)*(32*n^2-64*n+39)*a(n-1) +16*(4*n-5)*(4*n-7)*(2*n-3)*a(n-2)=0. - _R. J. Mathar_, Jun 07 2016
%t CoefficientList[Series[4 x/(16 x + Sqrt[2] Sqrt[Sqrt[1 - 16 x] + 1] Sqrt[1 - 16 x] - Sqrt[1 - 16 x] - 1), {x, 0, 40}], x] (* _Vincenzo Librandi_, Oct 31 2014 *)
%o (Maxima) a(n):=sum(2^i*binomial(2*n-i-1,n-i)*binomial(2*n+i-1,i),i,0,n);
%o (PARI) a(n)=sum(i=0,n,2^i*binomial(2*n-i-1,n-i)*binomial(2*n+i-1,i)) \\ _M. F. Hasler_, Oct 31 2014
%K nonn
%O 0,2
%A _Vladimir Kruchinin_, Oct 31 2014