{"id":"W4402831806","doi":"10.1080/00150517.2012.12428023","title":"Polynomial Forms for Alternating Sums of Products of Binomial-Catalan Numbers","year":2012,"lang":"en","type":"article","venue":"˜The œFibonacci quarterly","topic":"History and Theory of Mathematics","field":"Mathematics","cited_by":0,"is_retracted":false,"has_abstract":true,"ca_institutions":"Royal Military College of Canada; Royal Ottawa Mental Health Centre","funders":"","keywords":"Catalan number; Mathematics; Binomial (polynomial); Binomial coefficient; Catalan; Polynomial; Gaussian binomial coefficient; Combinatorics; Discrete mathematics; Pure mathematics; Negative binomial distribution; Statistics; Mathematical analysis; Linguistics; Poisson distribution","routes":{"ca_aff":true,"ca_fund":false,"ca_venue":false,"about_ca":false,"invisible_to_affiliation_only":false},"retraction":null,"screen":null,"direct_labels":[],"prediction":{"model_version":"metacan-v3-hybrid-931329e0061c","candidate_categories":[],"consensus_categories":[],"category_scores_codex":[0.001168296,0.0006575634,0.0004772773,0.001900763,0.001308016,0.003011778,0.0008081864,0.0007766277,0.006244481],"category_scores_gemma":[0.004869173,0.000324323,0.0006264541,0.001554112,0.002111093,0.003626634,0.001204975,0.001395979,0.001342914],"about_ca_system_candidate":false,"about_ca_system_consensus":false,"about_ca_system_score_codex":0.001150459,"about_ca_system_score_gemma":0.0004735524,"about_ca_topic_candidate":false,"about_ca_topic_consensus":false,"about_ca_topic_score_codex":0.0005637944,"about_ca_topic_score_gemma":0.0006644672,"domain_scores_codex":[0.9990335,0.0002600797,0.00005369915,0.0001404753,0.0003559132,0.0001563196],"domain_scores_gemma":[0.9988386,0.0004976377,0.0001669898,0.00007917136,0.0002757968,0.0001419554],"domain_codex":null,"domain_gemma":null,"domain_candidate":null,"domain_consensus":null,"study_design_codex":"theoretical_or_conceptual","study_design_gemma":"theoretical_or_conceptual","study_design_scores_codex":[0.0000155008,0.000008856093,0.0001370133,0.00002530577,0.000002585399,0.00008411912,0.0001736601,0.000897818,0.0008007268,0.9903087,0.000614094,0.006931563],"study_design_scores_gemma":[0.000007398538,0.00002524733,0.0002120977,0.00001956493,0.000006124439,0.0003663992,0.00008616872,0.009928871,0.0007900334,0.9818993,0.006640238,0.00001856395],"study_design_candidate":"theoretical_or_conceptual","study_design_consensus":"theoretical_or_conceptual","genre_codex":"empirical","genre_gemma":"empirical","genre_scores_codex":[0.4890911,0.002104722,0.3230172,0.001281569,0.0007130189,0.00007606565,0.000240222,0.000213527,0.1832626],"genre_scores_gemma":[0.9189678,0.001017091,0.04564708,0.0002808142,0.0006559744,0.00009980676,0.0002008544,0.0001218108,0.03300865],"genre_candidate":"empirical","genre_consensus":"empirical","teacher_disagreement_score":0.006244481,"threshold_uncertainty_score":0.02088988,"prediction_status":"machine_predicted_unvalidated"},"machine_scores":{"provisional":true,"baseline":true,"maturity_gate_passed":false,"score_opus":0.03885015306470962,"score_gpt":0.2944854149642005,"score_spread":0.2556352618994909,"validation_status":"score_only:v0-immature-baseline","note":"Baseline scores from an immature model (maturity gate not passed). Scores rank; they never assert a category."}}