Bibliographic record
Abstract
Differentiating $x^n$ by $n$ times gives $n!$, but what is the explicit function that connects the two? This article offers the unique insight on how the polynomial $x^n$ is found inside the series that expresses the factorial $n!$, i.e. $n!=f(x^n)$. Subtraction trees are the mathematical mechanism used to establish this connection. The process is here applied to powers $n=2 \to 6$, but this can be extended to any power $n$. Proofs using the mathematical method of induction are provided for each power, resulting in the respective function expression. Moreover, reworking this new function $n!=f(x^n)$ enable the determination of all the coefficients $^nC_k$ in a row $n$ of the Pascal's triangle (a worked example is provided). A Matlab/Octave program to compute this is enclosed for practical classroom activities.
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How this classification was reachedexpand
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 | 0.002 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.000 | 0.000 |
| 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 itClassification
machine, unvalidatedMachine predicted; a candidate call from one teacher head, 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".