{"id":"e-ef1be62a07232bc2","operation":{"map":"A013609","operation":"basis_map"},"operation_detail":"A013609","conclusion_id":"n-c840071f7c42bf18","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":"e2c52af5b236554dfa0b937e17f1d943d6fb80daf02077be33d0979cc9b888c5","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"basis_map","certificate_raw_size":16217,"certificate_compressed_size":3129,"conclusion_name":"basis_map_A013609 of Triangle read by rows. T(n, k) = (1/2) * binomial(2*(n - k + 1), n - k + 1) * binomial(2*n - k, k - 1) for n > 0, T(0, 0) = 1.","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-0a5e54eacdb24124","role":"sequence","name":"Triangle read by rows. T(n, k) = (1/2) * binomial(2*(n - k + 1), n - k + 1) * binomial(2*n - k, k - 1) for n > 0, T(0, 0) = 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-c840071f7c42bf18","claim":"real_rooted"},{"node_id":"n-c840071f7c42bf18","claim":"one_signed_coefficients"},{"node_id":"n-c840071f7c42bf18","claim":"hadamard_kernel"},{"node_id":"n-c840071f7c42bf18","claim":"interlacing"}]}