{"id":"e-d8fa1582574a2976","operation":{"map":"A007318","operation":"basis_map"},"operation_detail":"A007318","conclusion_id":"n-123dac0e8ec7cfb1","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":"9e7e9f721610c7bfd08d9c5bc473a6ef76327bb4f5d8d8cb60bb1517bae382a1","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"basis_map","certificate_raw_size":14909,"certificate_compressed_size":2948,"conclusion_name":"shift_x_plus_one of T(n,k) = binomial(n,2*k)*binomial(2*k,k) for 0 <= k <= n, triangle read by rows.","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-4b168caf0bc63f8b","role":"sequence","name":"T(n,k) = binomial(n,2*k)*binomial(2*k,k) for 0 <= k <= n, triangle read by rows.","uses":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]"}],"cited_by":[{"node_id":"n-123dac0e8ec7cfb1","claim":"one_signed_coefficients"},{"node_id":"n-123dac0e8ec7cfb1","claim":"hadamard_kernel"},{"node_id":"n-123dac0e8ec7cfb1","claim":"real_rooted"},{"node_id":"n-123dac0e8ec7cfb1","claim":"interlacing"}]}