%I #14 Nov 17 2023 04:43:13
%S 1,1,7,9,41,61,225,369,1195,2101,6227,11529,32059,61741,163727,325089,
%T 831505,1690981,4206145,8717049,21215481,44633821,106782837,227363409,
%U 536618341,1153594261,2693492305,5835080169,13507578125,29443836301
%N Expansion of d/dx log(1/(1-x/sqrt(1-4*x^2))).
%H Vincenzo Librandi, <a href="/A189053/b189053.txt">Table of n, a(n) for n = 0..100</a>
%F G.f. sqrt(1-4*x^2)/(16*x^4+sqrt(1-4*x^2)*(4*x^3-x)-8*x^2+1),
%F a(n)=n*sum(k=1..n, (binomial((n-2)/2,(n-k)/2)*2^(n-k-1)*((-1)^(n-k)+1))/k);
%F Conjecture: n*a(n) +(n-1)*a(n-1) +(-13*n+12)*a(n-2) +(-13*n+25)*a(n-3) +4*(14*n-27) *a(n-4) +4*(14*n-41)*a(n-5) +80*(-n+3)*a(n-6) +80*(-n+4)*a(n-7)=0. - _R. J. Mathar_, Jun 14 2016
%F a(n) ~ 5^((n+1)/2). - _Vaclav Kotesovec_, Nov 17 2023
%o (Maxima) a(n):=n*sum((binomial((n-2)/2,(n-k)/2)*2^(n-k-1)*((-1)^(n-k)+1))/k,k,1,n);
%o (PARI) x='x+O('x^66); /* that many terms */
%o Vec(deriv(log(1/(1-x/sqrt(1-4*x^2))))) /* show terms */ /* Joerg Arndt, Apr 16 2011 */
%K nonn
%O 0,3
%A _Vladimir Kruchinin_, Apr 16 2011