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Using the integral representation of Bernoulli numbers I obtain formally the integral representation of the double summation $$ \sum_{k=1}^{\infty}\sum_{j=0}^{k}\binom{k}{j}\frac{B_{j+k+1}}{j+k+1}=2\int_0^\infty\frac{t}{e^{2 \pi t}-1}\frac{dt}{t^2+(1+t^2)^2}=0.069591059035995961110566767049... $$ So the alternative form of the question is $$ \int_0^\infty\frac{t}{e^{2 \pi t}-1}\frac{dt}{t^2+(1+t^2)^2}=\frac14+\frac{\ln\phi}{1-2\phi} $$