exponential generating function for bernoulli numbers

Question

How I can find exponential generating function for this sequence

$(2^n − 1) B_n,$

where $B_n$ is Bernoulli numbers

Answer 1

We refer back to the definition of the Bernoulli numbers. They are defined by the EGF $$\frac{x}{e^x-1} = \sum_{k=0}^\infty B_k \frac{x^k}{k!}.$$

The substitution $x\mapsto 2x$ gives us $$\frac{2x}{e^{2x}-1} = \sum_{k=0}^\infty B_k 2^k \frac{x^k}{k!}.$$

Then the EGF of the sequence $\left< (2^n-1)B_n\right>_{n\in\mathbb{N}}$ is just $$\frac{2x}{e^{2x}-1} - \frac{x}{e^{x}-1} = \frac{-x}{e^x+1}.$$

So the $n^\text{th}$ number of the sequence is the coefficient of $x^n/n!$ in the exponential expansion of the above EGF.