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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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.002 | 0.009 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.002 | 0.002 |
| Science and technology studies | 0.002 | 0.002 |
| Scholarly communication | 0.004 | 0.005 |
| Open science | 0.002 | 0.006 |
| Research integrity | 0.001 | 0.005 |
| Insufficient payload (model declined to judge) | 0.029 | 0.024 |
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 itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".