The Bernoulli numbers Bk are the coefficients in the Taylor series expansion of Tanh. When n is finite we get a finite size sum which tells us the number of ways m objects can be combined taken k times. But we limit ourselves to small k, say five or six.Below are the rational approximations of the Bernoulli numbers. And we limit ourselves to recursive values of x, so tanh(x), summed over the three quarks, is always nearly a bound value. But the uniform distribution is the sum of actions by all quarks. This is the condition when motions are encoded to minimize redundancy, the -pLogF(p) are about the same, where F means finite log. Just to emphasize, this uniform distribution of gluon motions is the combined result of the three quarks.
The value, tanx, becomes three power series, one for each root. The recursion generating r^-k is third order, so among the three are six possible modes of motion being maintained. Is this right? Well they list six independent properties.
How are they recursive?
Likely across multiple roots, as in r1^3 = r2^2 + r3 + r1^(-1), they intermingle sequentially. At this point I don't know enough.
| B0 | = | 1/1 | |
| B1 | = | 1/1 − 1/2 | |
| B2 | = | 1/1 − 3/2 + 2/3 | |
| B3 | = | 1/1 − 7/2 + 12/3 − 6/4 | |
| B4 | = | 1/1 − 15/2 + 50/3 − 60/4 + 24/5 |
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