Consider the Bernoulli Polynomials $B_n\in\mathbb{R}$ given as the coefficients of the series:
$$\frac{t}{e^t-1}=\sum\limits_{n=0}^{\infty}B_n\frac{t^n}{n!}$$ and the Bernoulli polynomials gven by the following expansion:
$$\frac{t e^{xt}}{e^t-1}=\sum\limits_{n=0}^{\infty}B_n(x)\frac{t^n}{n!}$$
It seems to be that the following relation holds, but I do not know why:
$$-B_{2k+2}\left(\frac{1}{2}\right)=B_{2k+2}\cdot \left( 1-\frac{1}{2^{2k+1}} \right), \forall k\in\mathbb{N}$$
I calculated a couple of examples and suprisingly the equation is satisfied by the first Bernoulli numbers and polynomials. Does anyone know how to prove it?
Best regards