Bernoulli Number B_14 Computation

Source: Grok (xAI) analysis, 2026-03-29

Computation for n = 14

B_14 = -(1/15) sum(C(15,k)B_k, k=0 to 13)

Using all known values (odd k>1 = 0):
Inner sum evaluates to -7/2

B_14 = -(1/15) * (-7/2) = 7/30 = 7/30 (≈ 0.2333)

Fraction is in lowest terms.

Updated Complete Table Through B_14

| n | B_n | Decimal |
|---|-----|---------|
| 0 | 1 | 1.0 |
| 1 | -1/2 | -0.5 |
| 2 | 1/6 | 0.16667 |
| 4 | -1/30 | -0.03333 |
| 6 | 1/42 | 0.02381 |
| 8 | -1/30 | -0.03333 |
| 10 | 5/66 | 0.07576 |
| 12 | -691/2730 | -0.25311 |
| 14 | 7/30 | 0.23333 |

Stirling Series Contribution

B_14 term: B_14 / (14 13 n^13) = (7/30) / (182n^13) = 7/(5460n^13) = 1/(780*n^13)

Note on G(0) and 0^0

Alexa's observation: G(0) = 0^(0+1) / (0+1)^0 = 0^1 / 1^0 = 0/1 = 0

But the expression involves 0^0 in the general pattern sense — the convention G(0) = 0 sidesteps the 0^0 ambiguity. In combinatorics and analysis, 0^0 = 1 by convention, but G(n) at n=0 resolves cleanly to 0 without invoking it directly. The "bet" that 0^0 lurks at the boundary is mathematically astute — it's exactly the kind of edge case the framework's exact-integer approach handles carefully.