New Sums Identities In Weighted Catalan Triangle With The Powers Of Generalized Fibonacci And Lucas Numbers.
Bibliographic record
Abstract
In this paper, we consider a generalized Catalan triangle de ned by km n 2n n k for positive integer m: Then we compute the weighted half binomial sums with the certain powers of generalized Fibonacci and Lucas numbers of the form n X k=0 2n n+ k km n X tk; where Xn either generalized Fibonacci or Lucas numbers, t and r are integers for 1 m 6: After we describe a general methodology to show how to compute the sums for further values of m. 1. Introduction Shapiro [6] derived the following triangle similar to Pascals triangle with entries given by Bn;k = k n 2n n k ; which called Catalan triangle because the Catalan numbers Cn = 1 n+1 2n n are the entries in the rst column. Shapiro derived sums identities from the Catalan triangle. For example, he gave the following identities: n X p=1 (Bn;p) 2 = C2n 1 and n X p=1 Bn;pBn+1;p = C2n: We also refer to [5] and references therein for other examples. 2000 Mathematics Subject Classi cation. 11B37. Key words and phrases. Catalan triangle, sums identites, partial binomial sum, recursions. 1 2 EMRAH KILIC AND AYNUR YALCINER The authors [4] gave also an alternative proof of the identities above and established the following identity: n X p=1 (pBn;p) 2 = (3n 2)C2(n 1): In a somewhat di¤erent from the Catalan triangle, K¬l¬c and Ionascu [2] derived the following result: for any a 2 C f0g ; n X p=1 2n n+ k a + a k = 1 an (a+ 1) 2n + (n+ 1)Cn: The authors also gave applications to the generalized Fibonacci and Lucas sequences, de ned by Un = AUn 1 + Un 2; Vn = AVn 1 + Vn 2; where U0 = 0; U1 = 1; and V0 = 2; V1 = A; respectively. The Binet forms are Un = n n and Vn = n + n where ; = (A p )=2 and = A + 4: For example, we recall one result from [2]: n X
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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 | 0.001 | 0.004 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.001 |
| Bibliometrics | 0.002 | 0.002 |
| Science and technology studies | 0.001 | 0.001 |
| Scholarly communication | 0.002 | 0.003 |
| Open science | 0.001 | 0.002 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.010 | 0.002 |
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".