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A signed variant of A309132.
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%I #10 Jul 16 2019 04:09:43

%S 1,1,1,4,1,9,1,16,27,25,1,36,1,49,-75,64,1,81,-1,100,49,121,-1,144,

%T 125,169,-243,196,1,225,-1,256,363,289,-1225,324,1,361,-169,400,1,441,

%U -1,484,135,529,-1,576,343,625,-867,676,1,729,-3025,784,361,841,-1

%N A signed variant of A309132.

%C See A309132 for background and conjectures.

%F a(n) = numerator(R(n+1)/4^mod(n,2)) with R(n) = n/(nB(n) + dB(n)/n) and nB(n) = numerator(B(n-1, 1/2)), dB(n) = denominator(B(n-1, 1/2)) where B(n, x) denotes the Bernoulli polynomials.

%F |a(2*n)| = A309132(2*n + 1) for n >= 0.

%F a(2*n+1) = (n + 1)^2 for n >= 0.

%p nB := n -> numer(bernoulli(n-1,1/2)): dB := n -> denom(bernoulli(n-1,1/2)):

%p R := n -> n/(nB(n) + dB(n)/n): a := n -> numer(R(n+1)/4^irem(n,2)):

%p seq(a(n), n=0..58);

%Y Cf. A309132, A157779, A157780.

%K sign

%O 0,4

%A _Peter Luschny_, Jul 15 2019