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(PARI) a(n) = 4^n * (1 - 4^n) * bernfrac(2*n) / (2*n) + 2*imag(polylog(-2*n, I)); \\ Michel Marcus, May 30 2018
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Array[4^#*(1 - 4^#) BernoulliB[2 #]/(2 #) + EulerE[2 #] &, 16] (* Michael De Vlieger, May 24 2018 *)
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a(n) = 4^n * (1 - 4^n) * bernoulliBernoulli(2*n) / (2*n) + EulerE(2*n).
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