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A212297 a(n) = denominator(1 + Sum_{k=1..n} n^2 / Product_{j=1..k} 4*j^2).

%I A212297 #38 Sep 27 2019 06:30:14
%S A212297 4,16,256,9216,196608,11796480,8493465600,554906419200,
%T A212297 426168129945600,138078474102374400,1227364214243328000,
%U A212297 26731992586219683840000,15397627729662537891840000,3469598781750625204961280000,8160496334677470482068930560000
%N A212297 a(n) = denominator(1 + Sum_{k=1..n} n^2 / Product_{j=1..k} 4*j^2).
%H A212297 Charles R Greathouse IV, <a href="/A212297/b212297.txt">Table of n, a(n) for n = 1..225</a>
%e A212297 r(n) = 5/4, 33/16, 869/256, 48449/9216, 1504375/196608, 124787549/11796480, ....
%e A212297 From _Petros Hadjicostas_, Sep 26 2019: (Start)
%e A212297 a(3) = denominator(1 + 3^2/(4*1^2) + 3^2/(4*1^2 * 4*2^2) + 3^2/(4*1^2 * 4*2^2 * 4*3^2)) = denominator(1 + 9/4 + 9/64 + 9/2304) = denominator(869/256) = 256.
%e A212297 a(4) = denominator(1 + 4^2/(4*1^2) + 4^2/(4*1^2 * 4*2^2) + 4^2/(4*1^2 * 4*2^2 * 4*3^2) + 4^2/(4*1^2 * 4*2^2 * 4*3^2 * 4*4^2)) = denominator(1 + 16/4 + 16/64 + 16/2304 + 16/147456) = denominator(48449/9216) = 9216.
%e A212297 (End)
%p A212297 a := n -> denom(1 + add(n^2 / mul(4*j^2, j=1..k), k=1..n)):
%p A212297 seq(a(n), n=1..15); # _Peter Luschny_, Sep 26 2019
%t A212297 G[n_] := Module[{N=1, D=1}, Sum[N*=2*k-1; D*=2*k; (n/D)^2, {k, 1, n}] + 1]; a[n_] := Denominator[G[n]]; Array[a, 15] (* _Jean-François Alcover_, Sep 05 2015, translated from PARI *)
%o A212297 (PARI) G(n)=my(N=1,D=1);sum(k=1,n,N*=2*k-1;D*=2*k;(n/D)^2)+1
%o A212297 a(n)=denominator(G(n))
%o A212297 vector(15, n, a(n))
%Y A212297 Numerators are A212296.
%K A212297 nonn,frac
%O A212297 1,1
%A A212297 _Charles R Greathouse IV_, Jul 02 2012
%E A212297 Redefinition according to the data by _Peter Luschny_, Sep 26 2019