The Akiyama-Tanigawa algorithm for Bernoulli numbers
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Bibliographic record
Abstract
A direct proof is given for Akiyama and Tanigawa's algorithm for computing Bernoulli numbers. The proof uses a closed formula for Bernoulli numbers expressed in terms of Stirling numbers. The outcome of the same algorithm with di#erent initial values is also briefly discussed. 1 The Algorithm In their study of values at non-positive integer arguments of multiple zeta functions, S. Akiyama and Y. Tanigawa [1] found as a special case an amusing algorithm for computing Bernoulli numbers in a manner similar to "Pascal's triangle" for binomial coe#cients. Their algorithm reads as follows: Start with the 0-th row 1, 1 2 , 1 3 , 1 4 , 1 5 , . . . and define the first row by 1 (1 - 1 2 ), 2 ( 1 2 - 1 3 ), 3 ( 1 3 - 1 4 ), . . . which produces the sequence 1 2 , 1 3 , 1 4 , . . . . Then define the next row by 1 ( 1 2 - 1 3 ), 2 ( 1 3 - 1 4 ), 3 ( 1 4 - 1 5 ), . . . , thus giving 1 6 , 1 6 , 3 20 , . . . as the second row. In general...
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Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.000 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.004 | 0.001 |
| Scholarly communication | 0.000 | 0.001 |
| Open science | 0.001 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from it