By now, you get the idea of what we are doing here. What about the sum of cubes? Is there a general formula that expresses this relation?
Expanding this result gives:
Note there is no term with n.
Call These Leading Terms the Bernoulli Numbers
Solve the equation for the sum of fourth powers:
With the following Wolfram Language code, we can visualize the first 100 Bernoulli numbers by color: Positive Bernoulli numbers are indicated by Blue, negative Bernoulli numbers by Red, and the slots where the Bernoulli number is 0 is indicated by Gray.
Ada’s original letter, mentioning the Bernoulli numbers