{"id":"e-501bbcb2ca148024","operation":{"map":"A013611","operation":"basis_map"},"operation_detail":"A013611","conclusion_id":"n-9ed8bdf5901d9584","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":"e76de6759c75edd22c9cc8a37bbdf3c998fa1f1a821f473d3f0b1d77f7297ecc","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"basis_map","certificate_raw_size":13675,"certificate_compressed_size":2560,"conclusion_name":"basis_map_A013611 of T(n,k) = binomial(n,k)*binomial(n+k,k), 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-4689fb1386d440ad","role":"sequence","name":"T(n,k) = binomial(n,k)*binomial(n+k,k), 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-9ed8bdf5901d9584","claim":"real_rooted"},{"node_id":"n-9ed8bdf5901d9584","claim":"one_signed_coefficients"},{"node_id":"n-9ed8bdf5901d9584","claim":"hadamard_kernel"},{"node_id":"n-9ed8bdf5901d9584","claim":"interlacing"}]}