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%I #33 Jun 06 2017 23:34:20
%S 1,3,27,135,2835,15309,168399,938223,42220035,239246865,2727414261,
%T 15620645313,359274842199,2072739474225,23984556773175,
%U 139110429284415,12937269923450595,75340571907153465,878973338916790425,5135054769461249325,120160281605393234205
%N Numerator of (3/4)^n * binomial(2*n,n).
%C By analytic continuation to the entire complex plane there exist regularized values for divergent sums:
%C Sum_{k>=0} a(k)/A046161(k) = -i/sqrt(2).
%C Sum_{k>=0} (-1)^k*a(k)/A046161(k) = 1/2.
%C Sum_{k>=0} (-1)^(k+1)*a(k)/A046161(k) = -1/2.
%H G. C. Greubel, <a href="/A285008/b285008.txt">Table of n, a(n) for n = 0..925</a>
%F a(n) = numerator of (-3)^n*sqrt(Pi)/(Gamma(1/2-n)*Gamma(1+n)).
%F From _Robert Israel_, Apr 07 2017: (Start)
%F a(n) = 3*(2*n-1)*a(n-1)/A000265(n) for n >= 1.
%F a(n) = 3^n*binomial(2n,n)/A001316(n). (End)
%p A[0]:= 1:
%p for n from 1 to 100 do A[n]:=3*(2*n-1)*2^padic:-ordp(n,2)/n*A[n-1] od:
%p seq(A[i],i=0..100); # _Robert Israel_, Apr 07 2017
%t Numerator[Table[(-3)^n*Sqrt[Pi]/(Gamma[1/2-n]*Gamma[1+n]), {n,0,20}]]
%o (PARI) for(n=0,10, print1(numerator((3/4)^n*binomial(2*n,n)), ", ")) \\ _G. C. Greubel_, Jun 06 2017
%Y Cf. A046161 (denominators).
%Y Cf. A000265, A001316.
%K nonn,frac
%O 0,2
%A _Ralf Steiner_, Apr 07 2017