{"id":"e-4cd45aa0a6518701","operation":{"exponent_offset":1,"exponent_slope":1,"factor":["1","1"],"operation":"row_factor"},"operation_detail":"factor [1+1], exponent 1 n + 1","conclusion_id":"n-b3bdd177c3ec2fb9","premise_count":1,"theorem":"a product of polynomials is real-rooted iff every factor is; its roots are the union of theirs","lean_route":null,"lean_declarations":[],"certificate_hash":"279dccca8959d2a20acbb32acbc853b8e7d96645690a1293a4eb18fb0b67c4ff","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"row_factor","certificate_raw_size":4811,"certificate_compressed_size":1444,"conclusion_name":"Triangle read by rows: T(n,k), for k=-n..n-1, is the scaled (by 2^n n!) probability that the sum of n uniform [-1, 1] variables is between k and k+1.","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-492f384fe03c9268","role":"sequence","name":"row_factor parent of Triangle read by rows: T(n,k), for k=-n..n-1, is the scaled (by 2^n n!) probability that the sum of n uniform [-1, 1] variables is between k and k+1.","uses":{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]"}],"cited_by":[{"node_id":"n-b3bdd177c3ec2fb9","claim":"real_rooted"},{"node_id":"n-b3bdd177c3ec2fb9","claim":"one_signed_coefficients"},{"node_id":"n-b3bdd177c3ec2fb9","claim":"hadamard_kernel"}]}