{"id":"e-f25539c5a8b7f259","operation":{"exponent_offset":0,"exponent_slope":1,"factor":["1","1"],"operation":"row_factor"},"operation_detail":"factor [1+1], exponent 1 n + 0","conclusion_id":"n-b3367e055aead099","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":"2128867e9e2cbff16cb1bac9f825c11ee0c7376f63f6d472a6c88356d8786176","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"row_factor","certificate_raw_size":4808,"certificate_compressed_size":1442,"conclusion_name":"Irregular triangle T(n, k) = [x^k]( p(n, x) ), where p(n, x) = (1/x)*(1-x)^n*(1+x)^(n+2)*( Sum_{j >= 0} j^(n+1)*x^j ), read by rows.","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-b3367e055aead099","claim":"real_rooted"},{"node_id":"n-b3367e055aead099","claim":"one_signed_coefficients"},{"node_id":"n-b3367e055aead099","claim":"hadamard_kernel"}]}