{"id":"W2076367270","doi":"10.1134/s1061920809040086","title":"Some identities and congruences involving a certain family of numbers","year":2009,"lang":"en","type":"article","venue":"Russian Journal of Mathematical Physics","topic":"Advanced Mathematical Identities","field":"Mathematics","cited_by":19,"is_retracted":false,"has_abstract":true,"ca_institutions":"University of Victoria","funders":"","keywords":"Congruence relation; Bernoulli number; Stirling numbers of the second kind; Mathematics; Stirling number; Stirling numbers of the first kind; Factorial; Euler's formula; Pure mathematics; Combinatorics; Mathematical analysis","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":"codex-gemma-dda1882f352a","candidate_categories":[],"consensus_categories":[],"category_scores_codex":[0.0006640241,0.0002659574,0.0009693137,0.0001162821,0.00008232697,0.00008792807,0.0003530935,0.00008983885,0.00003719064],"category_scores_gemma":[0.001044103,0.000207067,0.0002522499,0.0001770412,0.0004099349,0.0009720257,0.00006407387,0.0003145312,0.000009009824],"about_ca_system_candidate":false,"about_ca_system_consensus":false,"about_ca_system_score_codex":0.00004679287,"about_ca_system_score_gemma":0.00007409561,"about_ca_topic_candidate":false,"about_ca_topic_consensus":false,"about_ca_topic_score_codex":0.000001173248,"about_ca_topic_score_gemma":3.909992e-7,"domain_scores_codex":[0.997467,0.00008826034,0.001254308,0.0001536386,0.0007188118,0.000318005],"domain_scores_gemma":[0.9972584,0.001135833,0.0009694414,0.0002730573,0.0001666313,0.0001966198],"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":[0.00002047807,0.0002802198,0.00001944424,0.0006845105,0.00008152393,0.00002907529,0.001757288,0.000007378659,0.001952488,0.9944127,0.0003904156,0.0003645244],"study_design_scores_gemma":[0.0004696718,0.0002340069,0.0001279696,0.001002865,0.0001490301,0.0001089189,0.00211526,0.0002433244,0.002660237,0.992678,0.000006931821,0.0002038045],"study_design_candidate":"theoretical_or_conceptual","study_design_consensus":"theoretical_or_conceptual","genre_codex":"empirical","genre_gemma":"empirical","genre_scores_codex":[0.9040534,0.002619076,0.07414957,0.00105569,0.0003007932,0.0005150372,0.00001677943,0.00009675235,0.01719294],"genre_scores_gemma":[0.9243441,0.0002248228,0.07452147,0.00009014083,0.0003042995,0.000002275018,3.976385e-7,0.0000346514,0.0004778854],"genre_candidate":"empirical","genre_consensus":"empirical","teacher_disagreement_score":0.02029069,"threshold_uncertainty_score":0.8443947,"prediction_status":"machine_predicted_unvalidated"},"machine_scores":{"provisional":true,"baseline":true,"maturity_gate_passed":false,"score_opus":0.03920171447362413,"score_gpt":0.3158032713090105,"score_spread":0.2766015568353863,"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."}}