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%I #21 Sep 30 2019 05:47:30
%S 1,17,69,1113,17817,285297,1141213,18260633,1168681737,18699007017,
%T 74796032037,1196736992841,19147791938817,306364680039081,
%U 1225458720340365,19607339566855065,5019478929156305865,80311662878468159865,321246651514020383485,5139946424277661728785
%N a(n) = numerator( Sum_{j=0..n} (j!/(2^j*floor(j/2)!)^2)^2 ).
%C This sequence is a variant of the Landau constants when the normalized central binomial is replaced by the normalized swinging factorial.
%C (1) A277233(n)/4^A005187(n) are the Landau constants. These constants are defined as G(n) = Sum_{j=0..n} g(j)^2 with the normalized central binomial
%C g(n) = (2*n)! / (2^n*n!)^2 = A001790(n)/A046161(n).
%C (2) A327495(n)/4^A327492(n) are the rationals considered here. These numbers are defined as H(n) = Sum_{j=0..n} h(j)^2 with the normalized swinging factorial
%C h(n) = n! / (2^n*floor(n/2)!)^2 = A163590(n)/A327493(n).
%C (3) In particular, this means that we have the pure integer representations
%C A277233(n) = Sum_{k=0..n}(A001790(k)*(2^(A005187(n) - A005187(k))))^2;
%C A327495(n) = Sum_{k=0..n}(A163590(k)*(2^(A327492(n) - A327492(k))))^2.
%C (4) A163590 is the odd part of the swinging factorial and A001790 is the odd part of the swinging factorial at even indices (see a comment from Aug 01 2009 in A001790). Similarly, A327493(2n)=A046161(2n) and A327493(2n+1) = 2*A046161(2n+1).
%C (5) A005187 are the partial sums of A001511, the 2-adic valuation of 2n, and A327492 are the partial sums of A327491.
%F Denominator(r(n)) = 4^A327492(n) = A327493(n)^2 = A327496(n).
%F a(n) = Sum_{k=0..n} (A163590(k)*(2^(A327492(n) - A327492(k))))^2.
%e r(n) = 1, 17/16, 69/64, 1113/1024, 17817/16384, 285297/262144, 1141213/1048576, 18260633/16777216, ...
%p A327495 := n -> numer(add(j!^2/(2^j*iquo(j,2)!)^4, j=0..n)):
%p seq(A327495(n), n=0..19);
%o (PARI) a(n)={ numerator(sum(j=0, n, (j!/(2^j*(j\2)!)^2)^2 )) } \\ _Andrew Howroyd_, Sep 28 2019
%Y Denominators in A327496.
%Y Cf. A327491, A327492, A327493, A327494.
%Y Cf. A277233, A005187, A056040, A000984, A001790/A046161, A163590, A001511.
%K frac,nonn
%O 0,2
%A _Peter Luschny_, Sep 27 2019
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