The Enchantress of numbers

Ada Lovelace , countess of Lovelace,daughter of the English poet Lord Byron and an amateur mathematician, is now widely known as the first computer programmer. What is amazing is the fact that she wrote a completely bug free program without ever running it on the machine.

Lovelace was deeply interested in Charles Babbage 's Analytic Engine and proposed a program to compute the Bernoulli numbers using punch cards. Her original note on the program describes a clever way of computing these numbers. Can you recreate the first computer program or make your own so as to efficiently find these numbers?

The $n$ th Bernoulli number $B_{n}$ is given by the double sum

$B_n=\sum _{ k=0 }^n{ \frac{1}{ k+1 }}\sum _{ r=0 }^k(-1)^r\dbinom{k}{r}r^n$

If $S_{500}=\sum _{ n=0 }^{ 500 }{ { B }_{ n } }$ can be represented as $-\frac{a}{b}$ for two co-prime integers $a$ and $b$ . Find the last five digits of $a+b$

Details and assumptions

• $B_{n}$ is the "original" Bernoulli number where $B_{1} = \frac{1}{2}$ not $\frac{-1}{2}$ .

Explicit example

$S_{5}=\sum _{ n=0 }^{ 5 }{ { B }_{ n } } = \frac{49}{30}$