Here's another way to get at the answer. You think you have a sequence of rationals that may be familiar:

1, 1/2,1/12,0,-1/720,...

The denominators seem more interesting than the numerator, so maybe the "right" sequence is:

1,2,12,1,720,...

You go to Sloane's Encyclopedia and enter the sequence, to no avail. You could now try superseeker, which looks at many transformations of the sequence, but for this few terms that will return too many hits. Let's try the one transformation you mentioned, and look at the exponential generating function, whose coefficients have denominators:

1, 2, 6, 1, 30, ...

Sloane's immediately identifies that sequence as the denominators of Bernoulli numbers, giving not only the generating function you started with but many other interesting factoids and references.

Here's another way to get at the answer. You think you have a sequence of rationals that may be familiar:

1, 1/2,1/12,0,-1/720,...

The denominators seem more interesting than the numerator, so maybe the "right" sequence is:

1,2,12,1,720,...

You go to Sloane's Encyclopedia and enter the sequence, to no avail. You could now try superseeker, which looks at many transformations of the sequence, but for this few terms that will return too many hits. Let's try the one transformation you mentioned, and look at the exponential generating function, whose coefficients have denominators:

1, 2, 6, 1, 30, ...

Sloane's immediately identifies that sequence as the denominators of Bernoulli numbers, giving not only the generating function you started with but many other interesting factoids and references.