{"id":"e-830e82d9cb2159c9","operation":{"map":"A013610","operation":"basis_map"},"operation_detail":"A013610","conclusion_id":"n-0515531ee2138839","premise_count":1,"theorem":"geometric family P_k = g(t) (a + d t)^k: T(p) = g(t) p(a + d t), so the roots of T(p) are those of g and the points (r - a)/d for the roots r of p (elementary)","lean_route":null,"lean_declarations":[],"certificate_hash":"a391cee560f3fcd48c9b76bc948abe7a2c1f962df8606c97d56395473e64bc0d","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"basis_map","certificate_raw_size":14522,"certificate_compressed_size":2901,"conclusion_name":"reversal of Riordan array (1, 3+3x).","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-1fe6eede339bc6ea","role":"sequence","name":"Triangle read by rows: T(n,k) is the number of ternary words (i.e., finite sequences of 0's, 1's and 2's) of length n having k occurrences of 01's (0 <= k <= floor(n/2)).","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-0515531ee2138839","claim":"real_rooted"},{"node_id":"n-0515531ee2138839","claim":"one_signed_coefficients"},{"node_id":"n-0515531ee2138839","claim":"hadamard_kernel"},{"node_id":"n-0515531ee2138839","claim":"interlacing"}]}