This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A285389 #70 Jun 27 2025 10:53:21
%S A285389 1,32,32768,268435456,70368744177664,295147905179352825856,
%T A285389 19807040628566084398385987584,10633823966279326983230456482242756608,
%U A285389 365375409332725729550921208179070754913983135744,50216813883093446110686315385661331328818843555712276103168
%N A285389 Denominator of Sum_{k=0..n^2-1} (-1)^k*sqrt(Pi)/(Gamma(1/2-k)*Gamma(1+k))/n.
%C A285389 All terms are powers of 2.
%C A285389 Lim_{n->inf} A285388(n)/a(n) = 2/sqrt(Pi).
%H A285389 Indranil Ghosh, <a href="/A285389/b285389.txt">Table of n, a(n) for n = 1..40</a>
%F A285389 a(n) is the denominator of Sum_{k=0..n^2-1} (binomial(2k,k)/4^k)/n.
%F A285389 a(n) = A000079(A285406(n)).
%F A285389 a(n) = denominator of n*binomial(2*n^2, n^2)/2^(2*n^2 -1). - _Ralf Steiner_, Apr 22 2017
%t A285389 Table[Denominator[Sum[Binomial[2k,k]/4^k,{k,0,n^2-1}]/n],{n,1,10}]
%t A285389 Denominator[Table[2^(1-2 n^2) n Binomial[2 n^2, n^2], {n, 1, 10}]] (* _Ralf Steiner_, Apr 22 2017 *)
%o A285389 (Python)
%o A285389 from sympy import binomial, Integer
%o A285389 def a(n): return (Integer(2)**(1-2*n**2) * Integer(n) * binomial(2*n**2, n**2)).denominator # _Indranil Ghosh_, Apr 27 2017
%o A285389 (Magma) [Denominator( n*(n^2+1)*Catalan(n^2)/2^(2*n^2-1) ): n in [1..21]]; // _G. C. Greubel_, Dec 11 2021
%o A285389 (Sage)
%o A285389 def A285389(n): return 2^(2*n^2 - 1 - (n^2).popcount() - valuation(n, 2))
%o A285389 [A285389(n) for n in (1..20)] # _G. C. Greubel_, Dec 12 2021
%Y A285389 Cf. A000079 (powers of 2), A285388 (numerators), A285406 (log base 2; A281264 + A007814), A190732 (2/sqrt(Pi)).
%Y A285389 Cf. A000120, A007814, A285406.
%K A285389 nonn,frac
%O A285389 1,2
%A A285389 _Ralf Steiner_, Apr 18 2017